Abstract
We define and develop an interior partial regularity theory for intrinsic energy minimising fractional harmonic maps from Euclidean space into smooth compact Riemannian manifolds for fractional powers strictly between zero and one. Intrinsic fractional harmonic maps are critical points of an energy whose first variation is a Dirichlet to Neumann map for the harmonic map problem on a half-space with a Riemannian metric which can degenerate/become singular along the boundary, depending on the fractional power. Similarly to the approach used to prove regularity for stationary intrinsic semi-harmonic maps, we take advantage of the connection between fractional harmonic maps and free boundary problems for harmonic maps in order to develop a partial regularity theory for the fractional harmonic maps we consider. In particular, we prove partial regularity for locally minimising harmonic maps with (partially) free boundary data on half-spaces with the aforementioned metrics up to the boundary; fractional harmonic maps then inherit this regularity. As a by-product of our methods we shed some new light on the monotonicity of the average energy of solutions of the degenerate linear elliptic equation related to fractional harmonic functions.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Harmonic maps are critical points of the Dirichlet energy for mappings of Riemannian manifolds and are non-linear analogues of solutions to Laplace’s equation. They play an important role in geometry and one of the most fundamental aspects of the theory of harmonic maps, which is intimately connected to the geometry and topology of the codomain manifold N, is their regularity. For maps with domains of dimension 2, the Dirichlet energy is invariant under conformal transformations of the domain; Hélein proved that harmonic maps are smooth in this case [21]. The fact that harmonic maps are smooth on domains of dimension two is also now a consequence of a result of Rivière [34] asserting the continuity of critical points of functionals with conformally invariant Lagrangian on two dimensional domains. Motivated by Rivière’s regularity result, at least partially, Da Lio and Rivière [8] introduced the notion of fractional harmonic mappings of manifolds; on a domain of dimension one these maps are critical points of functionals which satisfy a type of conformal invariance. Such maps generalise the notion of both harmonic mappings of manifolds and fractional harmonic functions (maps u with \((-{\varDelta })^su = 0\)) and Da Lio and Rivière showed the maps they considered are smooth on domains of one dimension.
Since the results of Da Lio and Rivière, fractional harmonic mappings of manifolds have been increasingly studied in recent years, particularly pertaining to their regularity. The purpose of this article is to extend the notion of a type of fractional harmonic map introduced by Moser [30]. In particular, we generalise the notion of intrinsic \(\frac{1}{2}\)-harmonic maps to all powers in (0, 1) and consider the regularity of a class of such maps. In contrast to other types of fractional harmonic maps, the maps we consider are intrinsic meaning that they are derived from an energy which only depends on the geometry of the target N and not on the embedding of N into Euclidean space. As we will discuss in more detail subsequently, we take advantage of the connection between free boundary harmonic maps and fractional harmonic maps to establish regularity for the latter. The results we obtain and methods we use lay the foundation for future studies into the connection between the regularity and singularities of intrinsic fractional harmonic maps, free boundary harmonic maps and the geometric and topological properties of N.
In order to motivate our definition of fractional harmonic mappings of manifolds, we first outline the pertinent aspects of the theory of fractional harmonic functions from Euclidean space into \({\mathbb {R}}\). Let \(m \in {\mathbb {N}}\) and \({\mathbb {R}}^{m+1}_+ = {\mathbb {R}}^m \times (0,\infty )\). Caffarelli and Silvestre [2] established that, for given boundary data \(u{:}\,\partial {\mathbb {R}}^{m+1}_+ \rightarrow {\mathbb {R}}\), solutions \(v{:}\,{\mathbb {R}}^{m+1}_+ \rightarrow {\mathbb {R}}\) of the Dirichlet problem:
satisfy \((-{\varDelta })^{\frac{1-\beta }{2}}u = {\partial ^{\beta }_{m+1}v} := - ( x_{m+1}^{\beta }\partial _{m+1}v)\vert _{\partial {\mathbb {R}}^{m+1}_+}\), where \((-{\varDelta })^{s}\) is the fractional Laplace operator of order \(s \in (0,1)\). One way to prove this result is to establish that
where \(E^{\beta }(v):= \frac{1}{2}\int _{ {\mathbb {R}}^{m+1}_+ }x_{m+1}^{\beta }\vert \nabla v\vert ^2\mathrm {d}x\), \(s = \frac{1-\beta }{2}\), \(C = C(m,s)\) and the infimum is taken over a suitable Sobolev space. We conclude that the first variation of the preceding functionals, respectively \(u \mapsto (-{\varDelta })^{\frac{1-\beta }{2}}u\) and \(u \mapsto \partial _{m+1}^{\beta }v\), must therefore coincide. Furthermore, if we wish to study u with \((-{\varDelta })^su = 0\), then we may equivalently consider minimisers (or even critical points) of \(E^{\beta }\) which satisfy \(v\vert _{\partial {\mathbb {R}}^{m+1}_+} = u\) and \( {\partial ^{\beta }_{m+1}v} =0\). It further follows from (1.2) that fractional harmonic functions \(u{:}\,{\mathcal {O}}\rightarrow {\mathbb {R}}\), where \({\mathcal {O}}\subset \partial {\mathbb {R}}^{m+1}_+\) is a domain, may reasonably be defined as critical points of either \(\vert \vert (-{\varDelta })^{\frac{s}{2}}u\vert \vert _{L^2({\mathcal {O}})}^2\) or \(\inf \{E^{\beta }(v){:}\,v\vert _{{\mathcal {O}}} = u \}\) (which are no longer equivalent). In order to define intrinsic fractional harmonic maps, we will work with a modification of the latter energy. For \(u{:}\,{\mathcal {O}} \rightarrow {\mathbb {R}}\), the first variation of \(\inf \{E^{\beta }(v){:}\,v\vert _{{\mathcal {O}}} = u \}\) is the Dirichlet to Neumann map \(u \mapsto -(x_{m+1}^{\beta }\partial _{m+1}v)\vert _{ {\mathcal {O}}}\) but we may no longer identify this with a fractional Laplace operator. Moreover, v satisfies (1.1) but with \(v\vert _{{\mathcal {O}}} = u\).
We generalise the notion of fractional harmonic function to mappings \(u{:}\,{\mathcal {O}} \rightarrow N\) in the following way, where \({\mathcal {O}} \subset \partial {\mathbb {R}}^{m+1}_+\) is an open domain but not necessarily all of \({\mathbb {R}}^{m+1}_+ \) and N is a smooth compact Riemannian manifold. For technical reasons, we assume henceforth that N is isometrically embedded in \({\mathbb {R}}^n\) for some n, which can always be achieved for smooth compact N by the theorem of Nash [31]. The forthcoming definition will be made rigorous at the beginning of Sect. 3. Define
where \(\beta \in (-1,1)\), \({\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N):= \{v \in {\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;{\mathbb {R}}^n){:}\,{v}( {x}) \in N\ \text {for a.e}\ {x} \in {\mathbb {R}}^{m+1}_+ \}\) and \({\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;{\mathbb {R}}^n)\) is the homogeneous Sobolev space with the square root of \(E^{\beta }\) as the norm. If we set \({\mathcal {O}} = \partial {\mathbb {R}}^{m+1}_+\) and N is replaced by \({\mathbb {R}}\) then \(I^{\beta }\) reduces to the energies considered by Caffarelli and Silvestre. Moreover, \(I^{\beta }\) is independent of the choice of embedding of N into Euclidean space since \(E^{\beta }\) is invariant under isometries of N. Hence we define critical points of \(I^{\beta }\) as intrinsic \(\frac{1-\beta }{2}\)-harmonic maps.
Our main result regarding fractional harmonic maps, see Theorem 3.8 in Sect. 3, asserts:
Theorem 1.1
Fix \({\mathcal {O}}\subset \partial {\mathbb {R}}^{m+1}_+\). When \(m \ge 3\) let \(\beta \in (-1,1)\) and when \(m =2\) let \(\beta \in (-3^{-1},1)\). Suppose \(u{:}\,{\mathcal {O}}\rightarrow N\) is a \(\frac{1-\beta }{2}\)-harmonic map that locally minimises \(I^{\beta }\) in \({\mathcal {O}}\). Then u is smooth with the possible exception of a set of points with vanishing \(m+\beta - 1\)-dimensional Hausdorff measure.
We observe that the domain of \(I^{\beta }\) plays a role in the regularity theory. We will take the domain of \(I^{\beta }\), again see Sect. 3 for details, to be the image of \({\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N)\) under some trace operator. The domain we choose does not necessarily coincide with the fractional Sobolev space \(H^{\frac{1-\beta }{2}}({\mathcal {O}};N)\); there are circumstances in which it is possible to construct \(H^{\frac{1-\beta }{2}}({\mathcal {O}};N)\) maps which do not have extensions to \({\mathbb {R}}^{m+1}_+\). This can happen for topological reasons, for example, such as when N is not simply connected [1]. We further note that the bound on the Hausdorff dimension of the singular set stated in the theorem may not be optimal; the given dimension is that which one would conjecture for u which are critical points of \(I^{\beta }\) with respect to inner and outer variations and is consistent with the scale-invariance of the problem. We expect that the dimension bound on the singular set can be improved using our constructions in Sect. 4 and similar arguments to Chapter 3 of [39] for example. It is also possible that our choice to minimise over \(v \in {\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N)\) may preclude some singularities, for example the map \(v = \vert (x',x_{m+1}) \vert ^{-1}(x',x_{m+1})\) is not an admissible extension of \(u = \vert x'\vert ^{-1} x'\) as it does not have finite \(E^{\beta }\) energy. However, the arguments we use are local and we expect our methods to apply to modifications of \(I^{\beta }\) which allow extensions v with only locally finite \(E^{\beta }\) energy. The restriction \(\beta > - 3^{-1}\) when \(m = 2\) is of a technical nature and results from the use of a Sobolev embedding theorem to identify elements of \({\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;{\mathbb {R}}^n)\) with functions, so that we may require they take values in N almost everywhere. It is possible that this restriction can be lifted with the use of an appropriate embedding theorem, we are not aware of such a result. An alternative would be to replace \({\dot{W}}^{1,2}_{\beta }\) with a Sobolev space of measurable functions from the outset (as opposed to representatives of Cauchy sequences). We further note that we do not consider the case \(m = 1\) and will indicate our reasons subsequently.
To prove Theorem 1.1 we will take advantage of the connection between the extensions \(v{:}\,{\mathbb {R}}^{m+1}_+\rightarrow N\) of fractional harmonic \(u{:}\,{\mathcal {O}} \rightarrow N\) and free boundary harmonic maps from \({\mathbb {R}}^{m+1}_+\) to N, similarly to Moser [30]. Moser observed that whenever \(I^0\) is differentiable at u, its first variation is the Dirichlet to Neumann map \({\varLambda }{:}\,u\mapsto (-\partial _{m+1}v)\vert _{{\mathcal {O}}}\) where \(v{:}\,{\mathbb {R}}^{m+1}_+\rightarrow N\) is a harmonic map with \(I^0(u) = E^0(v)\). Hence, intrinsic \(\frac{1}{2}\)-harmonic maps satisfy \({\varLambda }u = (-\partial _{m+1}v)\vert _{{\mathcal {O}}} = 0\), which may be interpreted as a zero Neumann condition on \({\mathcal {O}}\) for any v which minimises \(E^0\) among maps with u as boundary values on \({\mathcal {O}}\). Using this observation Moser established (partial) regularity for stationary \(\frac{1}{2}\)-harmonic \(u{:}\,{\mathcal {O}} \rightarrow N\), which are critical points of \(I^0\) with respect to inner and outer variations, using regularity theory for stationary free boundary harmonic maps \(v{:}\,{\mathbb {R}}^{m+1}_+\rightarrow N\) up to the free boundary \({\mathcal {O}}\). He showed the maps he considered are smooth when \(m = 1\) and smooth with the possible exception of a set of vanishing Hausdorff dimension \(m - 1\) when \(m \ge 2\). The following theorem is an abridged version of Theorem 4.3 in Sect. 4 which is our main theorem regarding the extensions v of fractional harmonic u.
Theorem 1.2
Suppose \(u{:}\,{\mathcal {O}} \rightarrow N\) locally minimises \(I^{\beta }\) and suppose v satisfies \(I^{\beta }(u) = E^{\beta }(v)\). There exists sets \({\varSigma }_{\text {int}}\subset {\mathbb {R}}^{m+1}_+\) and \({\varSigma }_{\text {bdry}} \subset {\mathcal {O}}\) such that the following holds. The set \({\varSigma }_{\text {int}}\) is relatively closed in \({\mathbb {R}}^{m+1}_+\) and has Hausdorff dimension at most \(m-2\). The set \({\varSigma }_{\text {bdry}}\) is relatively closed in \({\mathcal {O}}\) and has vanishing \(m+\beta -1\)-dimensional Hausdorff measure. The set \({\varSigma }:= {\varSigma }_{\text {int}} \cup {\varSigma }_{\text {bdry}}\) is relatively closed in \({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}\) and has vanishing \(m +\beta - 1\)-dimensional Hausdorff measure. Furthermore, we have \(v \in C^{\infty }({\mathbb {R}}^{m+1}_+{\backslash } {\varSigma }_{\text {int}};N)\), \(v \in C^{0,1}_{loc}(({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}){\backslash } {\varSigma };N)\) and for every multi-index \(\alpha ' \in {\mathbb {N}}^{m+1}\) with \(\alpha '_{m+1} = 0\) we have \(D^{\alpha '}v \in C^{0,1}_{loc}(({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}){\backslash } {\varSigma };{\mathbb {R}}^n)\).
Synonymously with our discussion following Theorem 1.1, we observe that the dimension bound on the singular set of v in \({\mathcal {O}}\) may not be optimal. The main purpose of this article is to give a proof of Theorem 1.2 which, to our knowledge, is new for all \(\beta \ne 0\). When \(\beta = 0\), existing theories actually improve the bound we have given on the dimension of the singular set in \({\mathcal {O}}\); this improvement comes from a dimension reduction argument as mentioned previously. This translates to a lower bound on the dimension of the singular set of u. In particular, as a consequence of the results of Hardt and Lin [18] and Duzaar and Steffen [10, 11], any \(v{:}\,{\mathbb {R}}^{m+1}_+ \rightarrow N\) with \(I^0(u) = E^{0}(v)\) are smooth up to \({\mathcal {O}}\) away from a set in \({\mathcal {O}}\) of Hausdorff dimension at most \(\text {dim}{\mathcal {O}} - 3\). They also give examples of minimisers with singular sets in \({\mathcal {O}}\) which have precisely the stated Hausdorff dimensions, thus showing that the dimension bounds for the singular set are optimal.
We would like to take advantage of Moser’s observation regarding the connection between fractional harmonic u and harmonic v with free boundary data in order to develop a theory for intrinsic fractional harmonic maps for powers in (0, 1). In \({\mathbb {R}}^{m+1}_+\) we are free to utilise the theory of harmonic mappings of manifolds to obtain the stated regularity in Theorem 1.2, provided the energies \(E^{\beta }\) correspond to the energy for harmonic mappings from \(({\mathbb {R}}^{m+1}_+,g)\) to N for some metric g. To see that this is the case and facilitate our subsequent analysis, we first recall the definition of harmonic maps for a class of conformally Euclidean metrics on \({\mathbb {R}}^{m+1}_+\). Let \(({\mathbb {R}}^{m+1}_+,g)\) be a smooth Riemannian manifold and suppose \(g=f(x_{m+1})\delta \), where \(\delta \) is the Euclidean metric, \(f{:}\,{\mathbb {R}}^{m+1}_+ \rightarrow (0,\infty )\) is smooth and \(x_{m+1}\) is the \(m+1\)th Euclidean coordinate. Let \(W^{1,2}({\varOmega };N) := \{v \in W^{1,2}({\varOmega };{\mathbb {R}}^n){:}\,v(x) \in N\ \text {for almost every}\ x \in {\varOmega }\}\), where \({\varOmega }\subset {\mathbb {R}}^{m+1}_+\) is open. If \(x_1,\ldots ,x_{m+1}\) are Euclidean coordinates on \({\varOmega }\), the energy density of \(v \in W^{1,2}({\varOmega };N)\) is given by \(e(v) = f^{-1}\sum _{i}\left\langle \frac{\partial v}{\partial x_i}, \frac{\partial v}{\partial x_i} \right\rangle := f^{-1} \langle \nabla v, \nabla v \rangle = f^{-1}\vert \nabla v\vert ^2\), where \(\langle \cdot , \cdot \rangle \) is the Euclidean inner product in \({\mathbb {R}}^n\). The energy of v, see [22], is defined as
Observe that when \(m = 1\), for this class of metric g we have \(E_{g} \equiv E_{\delta }\). The Euler–Lagrange equations for \(E_g\) have the form
where A is the second fundamental form of N, \(A(v)\left( \nabla v, \nabla v\right) := \sum _{i}A(v)\left( \partial _iv, \partial _i v\right) \) and \(\text {div},\nabla \) are the Euclidean divergence and gradient operators. Critical points of \(E_g\) in \(W^{1,2}({\varOmega };N)\) with respect to the dependent variable (weakly) satisfy (1.4) and are called (weakly) harmonic maps with respect to g.
When \(m \ge 2\) the Dirichlet energy \(E^{\beta }\) is the energy for maps \({\mathbb {R}}^{m+1}_+ \rightarrow N\) where \({\mathbb {R}}^{m+1}_+\) is equipped with a Riemannian metric \(x_{m+1}^{\alpha }\delta \); it follows from (1.3) that \(\beta = \frac{\alpha (m-1)}{2}\). When \(m = 1\), any metric of the form \(x_{m+1}^{\alpha }\delta \) gives rise to the energy \(E^0\) in view of (1.3). It is therefore not possible that \(E^{\beta }\) corresponds to the energy of mappings from \(({\mathbb {R}}^{m+1}_+, x_{m+1}^{\alpha }\delta )\) for \(\beta \ne 0\) when \(m = 1\) and, in this case, we would need a different approach. We do not consider this further in this article and hence assume \(m \ge 2\).
Near the boundary \({\mathcal {O}}\) we would like to apply the theory for free boundary harmonic mappings of Riemannian manifolds in order to obtain regularity for fractional harmonic maps defined on \({\mathcal {O}}\). However, for fractional powers \(\frac{1-\beta }{2} \in (0,1)\) with \(\beta \ne 0\) (i.e fractional powers in (0, 1) other than \(\frac{1}{2}\)) the metrics \(x^{\alpha }_{m+1} \delta \) may degenerate or become singular along the boundary depending on \(\beta = \frac{ \alpha (m-1)}{2}\). Known theory for free boundary harmonic maps is not applicable in this case and the main focus of this article will be to develop a regularity theory in this context.
Without geometric constraints on the target manifold N (such as requiring N has non-positive sectional curvature), harmonic maps and harmonic maps with free boundary data may exhibit discontinuities. For harmonic maps defined on a domain of dimension at least 3, free boundary data can have discontinuities on the boundary [11, 17, 18] and there are even harmonic maps into spheres which are discontinuous everywhere [33]. However, it is possible to establish (partial) regularity for harmonic maps which are minimising (or stationary) with respect to free boundary conditions. This is the primary reason that we consider minimisers.
The methodology we use to prove our theorem is based on existing theories for minimising (free boundary) harmonic maps. Similarly to these theories, our task is divided into two parts. First we prove Hölder continuity, and then that this continuity implies higher regularity. To establish Hölder continuity our main construction will be comparison maps. Such maps were first used in the theory of harmonic maps by Morrey in order to prove interior regularity of energy minimising harmonic maps on two dimensional domains [27]. Schoen and Uhlenbeck later showed that on domains of dimension at least three, minimising harmonic maps are regular away from a set of Hausdorff dimension at most the dimension of the domain minus three [37]; they also used manifold valued comparison maps, which are more difficult to construct on domains of dimension greater than two. The aforementioned results of Hardt and Lin and Duzaar and Steffen for minimising free boundary harmonic maps are established using methods based the constructions of Schoen and Uhlenbeck. We develop a (partial) generalisation of a construction of Luckhaus [23] as presented by Simon [39]; the advantage of this construction, at least in existing theories of harmonic maps, is that it can be used to show blow-up limits of sequences of minimisers are again locally minimising. We do not establish this fact here but expect similar results to follow from our constructions. In Sect. 4 we construct comparison maps on a half-cylinder and use these to construct comparison maps on half-balls in \({\mathbb {R}}^{m+1}_+\) centred on \(\partial {\mathbb {R}}^{m+1}_+\). We also establish monotonicity of a local, re-scaled version of \(E^{\beta }\) and estimates on the mean squared oscillation of the maps we consider via a blow-up argument. The combination of these results is enough to prove continuity. In order to prove Hölder continuity implies higher regularity (away from the singular set), we use a version of harmonic replacement to show \(\nabla v \in L^{\infty }\) similarly to [38]. We then establish energy decay for the derivatives of v via another blow-up argument which yields control of the mean squared oscillation of the derivatives of v tangential to \({\mathcal {O}}\).
To conclude our discussion, we compare the notions of intrinsic and extrinsic fractional harmonic maps and remark on known results regarding their regularity. The first to consider such maps were Da Lio and Rivière [8]. They analysed critical points u of the energy \(\vert \vert (-{\varDelta })^{\frac{1}{4}} {u}\vert \vert _{L^2(\partial {\mathbb {R}}^{2}_+ )}^2 \) in \({\dot{H}}^{\frac{1}{2}}\) under the constraint that u takes values in the unit \(n-1\) sphere \({\mathbb {S}}^{n-1}\) almost everywhere. In other words, they considered critical points of the functional
The maps Da Lio and Rivière considered fall into a larger class of maps known as extrinsic (fractional) harmonic maps. In contrast to intrinsic fractional harmonic maps, the energies for which extrinsic fractional harmonic maps are critical points depend upon the choice of embedding of the target manifold N into some Euclidean space. Moser [30] proposed to remove this dependence in the energy L so that only the geometry of N plays a role in the theory. In particular, to define \(I^0(u)\) he minimised \(E^0(v)\) over extensions \(v{:}\,{\mathbb {R}}^{m+1}_+\rightarrow N\) with \(v\vert _{{\mathcal {O}}}=u\) as opposed to minimising over extensions \(v{:}\,{\mathbb {R}}^{m+1}_+\rightarrow {\mathbb {R}}^n\) with \(v\vert _{{\mathcal {O}}}=u\) as is done to define L(u) (for \({\mathcal {O}} = \partial {\mathbb {R}}^2_+ \)). This choice results in another fundamental difference between intrinsic and extrinsic fractional harmonic maps which we illustrate for the power \(\frac{1}{2}\) (when \(\beta = 0\)). Observe that any \(v{:}\,{\mathbb {R}}^2_+\rightarrow {\mathbb {R}}^n\) for which \(L(u) = E^0(v)\) satisfies \({\varDelta }v = 0\) in \({\mathbb {R}}^{2}_+\) and is hence smooth on \({\mathbb {R}}^{2}_+\) (\({\varDelta }\) is the usual Laplace operator on Euclidean space). In contrast, any \(v{:}\,{\mathbb {R}}^{m+1}_+\rightarrow N\) for which \(I^0(u) = E^0(v)\) is a (weakly) harmonic map from \({\mathbb {R}}^{m+1}_+\) to N and satisfies
Unlike solutions of \({\varDelta }v = 0\), solutions of (1.5) may have singularities in \({\mathbb {R}}^{m+1}_+\) in general. We conclude extrinsic \(\frac{1}{2}\)-harmonic maps always have a smooth extension to \({\mathbb {R}}^{m+1}_+\) but intrinsic \(\frac{1}{2}\)-harmonic maps need not. The same is true for all powers in (0, 1).
The regularity theory for extrinsic \(\frac{1}{2}\)-harmonic maps has been extended to a range of fractional powers for maps into general target manifolds N by Da Lio et al. [6, 7, 9, 35, 36] etc, as well as Millot and Sire [25] for the power \(\frac{1}{2}\). Millot–Sire and Yu have also considered the regularity of extrinsic minimising fractional harmonic maps defined on the real line for powers in \( (0,\frac{1}{2})\) [26]. The methods Da Lio, Rivière and Schikorra used to obtain regularity take advantage of compensation phenomena, namely higher than expected regularity which can be derived from the Euler–Lagrange equations for the energies they considered. The higher regularity is revealed using a well chosen gauge transformation, in analogy to techniques used in the theory of harmonic maps. One of the earliest applications of this technique was used by Hélein to prove regularity for harmonic maps from Riemannian surfaces into spheres [20]. Millot, Sire and Yu instead consider the extensions of fractional harmonic maps and exploit their connection with free boundary problems for harmonic maps, a method first applied by Moser in the intrinsic case for \(\frac{1}{2}\)-harmonic maps to obtain (partial) regularity. To our knowledge, aside from the results we establish in this article, the only results regarding intrinsic fractional harmonic maps are those of Moser [30].
2 Preliminaries
The Euler–Lagrange equations for \(E^{\beta }\) at critical points \(v{:}\,{\mathbb {R}}^{m+1}_+ \rightarrow N\) are semi-linear with leading order term \(\text {div}(x_{m+1}^{\beta }\nabla v)\). We will require Sobolev spaces suited to the analysis of solutions of such equations, as well as the associated linear equation \(\text {div}(x_{m+1}^{\beta }\nabla v)=0\). For \(\beta \ne 0\) the coefficient \(x_{m+1}^{\beta }\) degenerates or becomes singular on \(\partial {\mathbb {R}}^{m+1}_+\) depending on the sign of \(\beta \) and the theory of uniformly elliptic second order partial differential equations does not apply on sets overlapping \(\partial {\mathbb {R}}^{m+1}_+\). Viewing the coefficient \(x_{m+1}^{\beta }\) as a weighting (density) of the Lebesgue measure \(\mathrm {d}x\), we may instead appeal to the theory of weighted second order degenerate elliptic equations. We recall and define the function spaces, and some of their analytical properties, necessary for our analysis and then record properties of solutions to \(\text {div}(x_{m+1}^{\beta }\nabla v)=0\) which we will require for the study of solutions of the Euler–Lagrange equations of \(E^{\beta }\).
For every \(\beta \in (-1,1)\), there exists \(C>0\) such that \( \frac{1}{\int _{B }\mathrm {d}x}\int _{B } \vert x_{m+1}\vert ^{\beta }\mathrm {d}x \frac{1}{\int _{B }\mathrm {d}x}\int _{B } \vert x_{m+1}\vert ^{-\beta }\mathrm {d}x \le C\) for every ball \(B \subset {\mathbb {R}}^{m+1}\), where \(\mathrm {d}x\) is the Lebesgue measure on \({\mathbb {R}}^{m+1}\). Hence \(\vert x_{m+1}\vert ^{\beta }\) is a weight of Muckenhoupt class \(A_2\), see [19] for an overview of these weights. Every such weight is canonically associated to corresponding Sobolev and Lebesgue spaces. Let \({\varOmega }\subset {\mathbb {R}}^{m+1}\). Define
which is a Hilbert space, see [5, Theorem 3.4.1], where the inner product of \(f,g \in L^2_{\beta }({\varOmega };{\mathbb {R}}^n)\) is given by \(\left\langle f, g\right\rangle _{L^2_{\beta }({\varOmega };{\mathbb {R}}^n)} = \int _{{\varOmega }}\left\langle f, g\right\rangle \vert x_{m+1}\vert ^{\beta }\mathrm {d}x\) where \(\left\langle f, g\right\rangle \) is the inner product of f and g in \({\mathbb {R}}^{n}\). Define
where \(\partial _iv\) denotes the weak partial derivative of v with respect to \(x_i\). Proposition 2.1.2 of [40] guarantees that \(W^{1,2}_{\beta }({\varOmega };{\mathbb {R}}^n)\) is a Hilbert space with the inner product \(\left\langle v, w\right\rangle _{W^{1,2}_{\beta }({\varOmega };{\mathbb {R}}^n)} = \int _{{\varOmega }}\left\langle v, w\right\rangle \vert x_{m+1}\vert ^{\beta }\mathrm {d}x + \int _{{\varOmega }}\left\langle \nabla u, \nabla v\right\rangle \vert x_{m+1}\vert ^{\beta }\mathrm {d}x\) for \(v,w \in W^{1,2}_{\beta }({\varOmega };{\mathbb {R}}^n)\), where we write \(\left\langle \nabla v, \nabla w\right\rangle = \sum _{i = 1}^{m+1}\left\langle \partial _iv, \partial _iw\right\rangle \). We also define the Hilbert space \({W}^{1,2}_{\beta ,0}({\varOmega };{\mathbb {R}}^n)\) as the closure of \(C^{\infty }_0({\varOmega };{\mathbb {R}}^n)\) in \({W}^{1,2}_{\beta }({\varOmega };{\mathbb {R}}^n)\) with respect to the norm induced by the inner product on \( {{W}^{1,2}_{\beta }({\varOmega };{\mathbb {R}}^n)}\). When \(\beta = 0\) we omit the subscript \(\beta \) from the preceding notation.
It is worth noting that, for every \(\beta \in (-1,1){\backslash }\{0\}\), approximation by smooth functions in \(L^2_{\beta }({\varOmega };{\mathbb {R}}^n)\) and \(W^{1,2}_{\beta }({\varOmega };{\mathbb {R}}^n)\) works in the same way as for the spaces \(L^2({\varOmega };{\mathbb {R}}^n)\) and \(W^{1,2}({\varOmega };{\mathbb {R}}^n)\). The details of this process are given in Theorem 2.1.4 and Corollary 2.1.5 in [40]. The spaces \(W^{1,2}_{\beta },W^{1,2}_{\beta ,0}\) and \(L^2_{\beta }\) also have essentially the same analytical properties, such as completeness, reflexivity etc. as the unweighted spaces \(W^{1,2},W^{1,2}_0\) and \(L^2\) respectively. Moreover, since \(\vert x_{m+1}\vert ^{\beta }\) is an \(A_2\) weight, many inequalities that hold for the spaces \(W^{1,p}\), such as the Poincaré inequality, have counterparts for functions in \(W^{1,2}_{\beta }\) [19].
We will need to refer to the relationship between \(W^{1,2}_{\beta }\) and \(W^{1,p}\) and record it in the following Lemma.
Lemma 2.1
Let \(\beta \in (-1,1)\) and suppose \({\varOmega }\subset {\mathbb {R}}^{m+1}_+\) is open and bounded.
-
1.
If \({\overline{{\varOmega }}} \subset {\mathbb {R}}^{m+1}_+\) then \(W^{1,2}_{\beta }({\varOmega };{\mathbb {R}}^n) = W^{1,2}({\varOmega };{\mathbb {R}}^n)\).
-
2.
If \(\beta \le 0\) then \(W^{1,2}_{\beta }({\varOmega };{\mathbb {R}}^n) \subset W^{1,2}({\varOmega };{\mathbb {R}}^n)\).
-
3.
If \(\beta >0\) then \(W^{1,2}_{\beta }({\varOmega };{\mathbb {R}}^n) \subset W^{1,p}({\varOmega };{\mathbb {R}}^n)\) for every \(1 \le p < \frac{2}{1+\beta }\).
Proof
Part 1 follows as in this case the norms on \({W^{1,2}_{\beta }({\varOmega };{\mathbb {R}}^n)}\) and \( {W^{1,2}({\varOmega };{\mathbb {R}}^n)}\) are equivalent. Part 2 follows as in this case the \({W^{1,2} ({\varOmega };{\mathbb {R}}^n)}\) norm is dominated by a constant multiple (depending on \({\varOmega }\)) of the \({W^{1,2}_{\beta }({\varOmega };{\mathbb {R}}^n)}\) norm. Part 3 follows from Hölder’s inequality. \(\square \)
2.1 Weighted homogeneous Sobolev spaces
Consider the Dirichlet energies
where \(\beta \in (-1,1)\), which are well defined on
The energy \(E^{\beta }\) is naturally associated to the following Sobolev space.
Definition 2.2
Let \(\beta \in (-1,1)\). The Weighted Homogeneous Sobolev Space \({\dot{W}}_{\beta }^{1,2}({\mathbb {R}}^{m+1}_+;{\mathbb {R}}^n)\) is the completion of \({\mathcal {D}}_+({\mathbb {R}}^{m+1}_+;{\mathbb {R}}^n)\) with respect to the metric induced by the square root of \(E^{\beta }\).
The elements of \({\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;{\mathbb {R}}^n)\) are, strictly speaking, equivalence classes of Cauchy sequences and it will be necessary to have tangible representatives of these classes which may take values in N.
Lemma 2.3
Let \(m \in {\mathbb {N}}\) with \(m \ge 2\) and \({\varOmega }\) be an open bounded subset of \({\mathbb {R}}^{m+1}_+\). If \(m = 2\) let \(\beta \in (-3^{-1},1)\) and if \(m \ge 3\) let \(\beta \in (-1,1)\). Then the restriction operator \(I{:}\,{\mathcal {D}}_+({\mathbb {R}}^{m+1}_+;{\mathbb {R}}^n) \rightarrow {W}^{1,2}_{\beta }({\varOmega };{\mathbb {R}}^n){:}\,f \mapsto f\vert _{{\varOmega }}\) extends to a bounded linear operator \(I{:}\,{\dot{W}}_{\beta }^{1,2}({\mathbb {R}}^{m+1}_+;{\mathbb {R}}^n) \rightarrow {W}^{1,2}_{\beta }({\varOmega };{\mathbb {R}}^n)\). Moreover,
for every \(v \in {\dot{W}}_{\beta }^{1,2}({\mathbb {R}}^{m+1}_+;{\mathbb {R}}^n) \) where C is a positive constant depending on m and \({\varOmega }\).
Proof
It suffices to establish that (2.2) holds for all \(\phi \in {\mathcal {D}}_+({\mathbb {R}}^{m+1}_+;{\mathbb {R}}^n)\) as this space is dense in \({\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;{\mathbb {R}}^n)\) and \({W}^{1,2}_{\beta }({\varOmega };{\mathbb {R}}^n)\) is a Banach Space. Let \(\phi \in {\mathcal {D}}_+({\mathbb {R}}^{m+1}_+;{\mathbb {R}}^n)\). By definition we have \(\vert \vert \nabla \phi \vert \vert ^2_{L^2_{\beta }({\varOmega };{\mathbb {R}}^n)} \le \vert \vert \phi \vert \vert ^2_{{\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;{\mathbb {R}}^n)}\) so we need to estimate \(\vert \vert \phi \vert \vert ^2_{L^2_{\beta }({\varOmega };{\mathbb {R}}^n)} \) in terms of \(\vert \vert \phi \vert \vert ^2_{{\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;{\mathbb {R}}^n)}\).
Suppose \(m \ge 3\). We have \(\phi (\cdot ,x_{m+1}) \in W^{1,2}({\mathbb {R}}^{m};{\mathbb {R}}^n)\) for every \(x_{m+1} \in [0,\infty )\). Let \(\nabla '\) denote the derivative of \(\phi \) with respect to \(x'\) for \((x',x_{m+1}) \in {\mathbb {R}}^m \times [0,\infty )\), define \(a = \inf _{{\varOmega }}(x_{m+1})\) and \(b = \sup _{{\varOmega }}(x_{m+1})\) and set \(l(x_{m+1}) = {\varOmega }\cap ({\mathbb {R}}^m \times \{x_{m+1}\})\). We apply Fubini’s Theorem, together with Hölder’s inequality, for conjugate exponents \(\frac{m}{m-2}\) and \(\frac{m}{2}\), and the Sobolev inequality in \({\mathbb {R}}^{m}\) to see that
We conclude that \(\vert \vert \phi \vert \vert ^2_{{W}^{1,2}_{\beta }({\varOmega };{\mathbb {R}}^n)} \le C\vert \vert \phi \vert \vert ^2_{{\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;{\mathbb {R}}^n)}\).
In order to apply the preceding method of proof when \(m = 2\) we would need to use the Sobolev Embedding Theorem on \({\mathbb {R}}\). However, in this case the theorem yields \(W^{1,2}({\mathbb {R}})\hookrightarrow L^\infty ({\mathbb {R}})\) and our previous method of proof is no longer viable. However, provided \(\beta > - 3^{-1}\), Corollary 2 in Section 2.1.7 of [24] implies that for every \(\phi \in C_0^{\infty }({\mathbb {R}}^3;{\mathbb {R}}^n)\), and therefore by approximation every \(\phi \in W^{1,2}_{\beta }({\mathbb {R}}^3;{\mathbb {R}}^n)\), we have
The even reflection with respect to \(\partial {\mathbb {R}}^{m+1}_+\), denoted \({\tilde{\phi }}\), of a \(\phi \in {\mathcal {D}}_+({\mathbb {R}}^{m+1}_+;{\mathbb {R}}^n)\) is in \(W^{1,2}_{\beta }({\mathbb {R}}^3;{\mathbb {R}}^n)\) and hence, applying Hölder’s inequality, we find
for every \( \phi \in {\mathcal {D}}_+({\mathbb {R}}^3_+;{\mathbb {R}}^n)\). We again conclude \(\vert \vert \phi \vert \vert ^2_{{W}^{1,2}_{\beta }({\varOmega };{\mathbb {R}}^n)} \le C\vert \vert \phi \vert \vert ^2_{{\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;{\mathbb {R}}^n)}\). \(\square \)
Remark 2.4
We can further embedd \({\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;{\mathbb {R}}^n)\) into \(W^{1,p}({\varOmega };{\mathbb {R}}^n)\), where p depends on \(\beta \), for open bounded \({\varOmega }\subset {\mathbb {R}}^{m+1}_+\) using Lemma 2.1.
2.2 Compactness of the embedding \(W^{1,2}_{\beta } \hookrightarrow L^2_{\beta }\)
We will require an analogue of the compact embedding \(W^{1,2}\hookrightarrow L^2\) in order to analyse re-scaled limits of bounded sequences of Sobolev functions. Away from the boundary, that is, for \({\varOmega }\) with \({\overline{{\varOmega }}}\subset {\mathbb {R}}^{m+1}_+\), the compactness of the inclusion \(W^{1,2}\hookrightarrow L^2\) yields the compactness of the inclusion \(W^{1,2}_{\beta } \hookrightarrow L^2_{\beta }\) in view of Lemma 2.1. We have not been able to find a proof of compactness near \(\partial {\mathbb {R}}^{m+1}_+\) in the literature so present one for completeness. Let \(B_r(y) = \{x \in {\mathbb {R}}^{m+1}{:}\,\vert x - y\vert <r\}\) and \(Q_r(y) = \{x \in {\mathbb {R}}^{m+1}{:}\,\vert x_i - y_i\vert <r, i = 1,\ldots , m+1\}\) and define \(B^+_r(y) = B_r(y)\cap {\mathbb {R}}^{m+1}_+\) and \(Q^+_r(y) = Q_r(y)\cap {\mathbb {R}}^{m+1}_+\) for \(y \in \partial {\mathbb {R}}^{m+1}_+\).
Lemma 2.5
Let \(r>0\), \(y \in \partial {\mathbb {R}}^{m+1}_+\) and suppose \((v_j)_{j \in {\mathbb {N}}}\) is a sequence in \( {W}^{1,2}_{\beta }({\varOmega };{\mathbb {R}}^n)\) which satisfies \(\sup _j \vert \vert v_j \vert \vert _{ {W}^{1,2}_{\beta }({\varOmega };{\mathbb {R}}^n)}< \infty \) where \({\varOmega }\) is either \(Q^+_r(y)\) or \(B^+_r(y)\). Then there exists a subsequence \((v_{j_k})_{k \in {\mathbb {N}}}\) and a \(v \in {W}^{1,2}_{\beta }({\varOmega };{\mathbb {R}}^n)\) such that
-
1.
\(v_{j_k} \rightharpoonup v\) in \({W}^{1,2}_{\beta }({\varOmega };{\mathbb {R}}^n)\)
-
2.
\(v_{j_k} \rightarrow v\) in \(L^2_{\beta }({\varOmega };{\mathbb {R}}^n)\)
-
3.
\(\int _{{\varOmega }}x_{m+1}^{\beta }\vert \nabla v \vert ^2\mathrm {d}x \le \liminf _{k \rightarrow \infty }\int _{{\varOmega }}x_{m+1}^{\beta }\vert \nabla v_{j_k} \vert ^2\mathrm {d}x\).
Proof
For \(\beta = 0\), a proof can be found in [39, Section 1.3, Lemma 1]. Otherwise, statement 1 follows from the weak sequential compactness of the unit ball in a Hilbert space and statement 3 follows from statement 2 and the lower semi-continuity of a Hilbert space norm. Hence, the main task is to prove statement 2. We may assume \(r = 1\) and \(y = 0\) since statement 2 is invariant under rescaling and translations with respect to \(x_i\) for \(i = 1,\ldots ,m\). We further assume that \({\varOmega }= Q^+_1(0)\) since if \({\varOmega }= B^+_1(0)\) and the result is true on \(Q^+_1(0)\) then we may compose with the bi-Lipschitz, piecewise \(C^1\) with piecewise \(C^1\) inverse map \(Q^+_1(0) \rightarrow B^+_1(0){:}\,x \mapsto \text {max}_i\vert x_i\vert \vert x\vert ^{-1}x\) and deduce the statement on \(B^+_1(0) \).
Suppose \((v_j)_{j \in {\mathbb {N}}}\) is a sequence with \(v_j \in {W}^{1,2}_{\beta }(Q^+_1(0);{\mathbb {R}}^n)\) for every j and which satisfies
for some positive constant M. Relabelling if necessary, suppose \((v_j)_{j \in {\mathbb {N}}}\) is also the subsequence which satisfies \(v_j \rightharpoonup v\) for \(v \in W^{1,2}_{\beta }(Q^+_1(0);{\mathbb {R}}^n)\). Then
Let \(Q' = (-1,1)^m\), define \( Q_i = \left\{ (x',x_{m+1}) \in Q^+_1(0){:}\,(i+1)^{-1} < x_{m+1} \le 1\right\} \) for \(i \in {\mathbb {N}}\) and let \({\hat{Q}}_i = Q^+_1(0){\backslash } Q_i = Q' \times [0,(i+1)^{-1}]\). In view of (2.3), we have \( \sup _{j \in {\mathbb {N}}}\vert \vert v_j\vert \vert _{W^{1,2}_{\beta }(Q_i;{\mathbb {R}}^n)} \le M\) for each \(i\in {\mathbb {N}}\). Hence, using the compactness of the embedding \(W^{1,2}_{\beta }(Q_1;{\mathbb {R}}^n) \hookrightarrow L^2_{\beta }(Q_1;{\mathbb {R}}^n)\), we find a \({\tilde{v}} \in W^{1,2}_{\beta }(Q_1;{\mathbb {R}}^n)\) and a subsequence, which we denote \((v_{j})_{j \in {\varLambda }_1}\) for an infinite set \({\varLambda }_1 \subset {\mathbb {N}}\), which satisfies \(v_{j} \rightharpoonup {\tilde{v}}\) in \(W^{1,2}_{\beta }(Q_1;{\mathbb {R}}^n)\), \(v_{j} \rightarrow {\tilde{v}}\) in \(L^2_{\beta }(Q_1;{\mathbb {R}}^n)\) and almost everywhere as \(j \rightarrow \infty \) with \(j \in {\varLambda }_1\). Notice that \((v_j)_{j \in {\varLambda }_1}\) converges weakly to v in \(W^{1,2}_{\beta }(Q_1;{\mathbb {R}}^n)\) because \((v_j)_{j \in {\mathbb {N}}}\) does and so, by the uniqueness of weak limits, we deduce \({\tilde{v}} = v\) in \(Q_1\). Hence \(v_{j} \rightarrow v\) in \(L^2_{\beta }(Q_1;{\mathbb {R}}^n)\) and almost everywhere as \(j\rightarrow \infty \) as well. Repeating this process inductively for every \(i \in {\mathbb {N}}\), we obtain sequences \((v_{j})_{j\in {\varLambda }_i}\) with \({\varLambda }_{i+1} \subset {\varLambda }_{i}\) such that \((v_{j})_{j \in {\varLambda }_i}\) converges to v in \(L^2_{\beta }(Q_i;{\mathbb {R}}^n)\) and almost everywhere in \(Q_i\). Hence we can choose an increasing sequence of numbers \((k_i)_{i \in {\mathbb {N}}}\) with \(k_i\in {\varLambda }_i\) such that
for \(k \ge k_i\). Then the sequence \((v_{k_i})_{i\in {\mathbb {N}}}\) converges to v almost everywhere in \(Q^+_1(0)\) and in \(L^2_{\beta }(Q_k;{\mathbb {R}}^n)\) for all \(k \in {\mathbb {N}}\) as \(i \rightarrow \infty \). Observe that
By (2.5) we have \(\int _{Q_{i}}x_{m+1}^{\beta }\vert v_{k_i} - v\vert ^2\mathrm {d}x < \frac{1}{i^{2+\beta }2^i} \rightarrow 0\) as \(i \rightarrow \infty \) so we consider the remaining term in (2.6). Using Chebychev’s inequality combined with Fubini’s Theorem, we may choose \(c_i \in \left( (i+1)^{-1},i^{-1}\right) \) such that
Now for each \(i \in {\mathbb {N}}\), we calculate
We apply Hölder’s inequality and (2.4) to see that
The bound for the integral on the left hand side of (2.9) but with v replaced by \(v_{k_i}\) is identical. We apply (2.7) followed by (2.5) to see that
Finally we combine (2.5), (2.6), (2.8), (2.9) and (2.10) and let \(i \rightarrow \infty \) to conclude the proof. \(\square \)
Remark 2.6
The method of proof of Lemma 2.5 is also valid for \({\varOmega }\) of the form \({\varOmega }= {\mathcal {O}} \times [0,r]\) for \(r>0\) and \({\mathcal {O}} \subset \partial {\mathbb {R}}^{m+1}_+\).
2.3 Energy decay for a linear Neumann-type problem
When examining the limit of re-scaled sequences of Sobolev functions, as in the proof of Lemma 4.18, we will obtain a weak solution of the Neumann-type problem
for some \(R>0\) and \(x_0 \in \partial {\mathbb {R}}^{m+1}_+\). The rate of decay of the re-scaled energy of such solutions on concentric half-balls centred at \(x_0\) will play a role in the proof of the aforementioned lemma and we estimate this decay here.
Lemma 2.7
Let \(\beta \in (-1,1)\), \(x_0 \in \partial {\mathbb {R}}^{m+1}_+\) and suppose \(v \in W^{1,2}_{\beta }(B^+_R(x_0);{\mathbb {R}}^n)\) satisfies
for every \(\psi \in C_0^{\infty }(B_R(x_0);{\mathbb {R}}^n)\). There exists a \(\gamma = \gamma (m,\beta ) \in (0,1)\) and a positive constant C such that
for every \(r \le \frac{R}{2}\).
Proof
The even reflection of v in \(\partial {\mathbb {R}}^{m+1}_+\), which we do not relabel, belongs to \(W^{1,2}_{\beta }(B_R(x_0);{\mathbb {R}}^n)\) and satisfies
for every \(\psi \in C_0^{\infty }(B_R(x_0);{\mathbb {R}}^n)\). A result of Fabes et al., see [13, Theorem 2.3.12], implies the local Hölder continuity of v in \(B_R(x_0)\). In particular, there exists a constant C such that
for some \(\gamma \in (0,1)\) and every \(x,y \in B_{ \frac{R}{2}}(x_0)\).
By approximation, (2.14) holds for every \(\psi \in W^{1,2}_{\beta ,0} (B_R(x_0);{\mathbb {R}}^n)\). Let \(\eta \in C^{\infty }_0(B_{r}(x_0);[0,1])\) be a cutoff function with \(\eta \equiv 1\) in \(B_{\frac{r}{2}}(x_0)\) and \(\vert \nabla \eta \vert \le \frac{C}{r}\) for a fixed positive \(C\ge 2\). We observe that \(\phi = \eta ^2(v-\lambda )\) is an admissible test function for every \(\lambda \in {\mathbb {R}}^n\). Testing (2.14) against \(\phi \) and using Young’s inequality, \(ab \le \delta \frac{a^2}{2} + \frac{b^2}{2\delta }\) for \(a,b \ge 0\) and \(\delta >0\), we see that
We choose \(\delta = \frac{1}{2}\), set \(\lambda = v(x_0)\), recall \(\vert \nabla \eta \vert \le \frac{C}{r}\) and apply (2.15) to see that
for another positive \(C>0\), independent of \(r \le \frac{R}{2}\). Multiplying (2.16) by \((2^{-1}r)^{1-m-\beta }\) concludes the proof. \(\square \)
2.4 Boundary monotonicity formula for the average energy of solutions to the degenerate linear equation
In order to utilise a version of the method of harmonic replacement, see Lemma 4.23 in Sect. 4.12, we need to know how the average energy of solutions to \(\text {div}(\vert x_{m+1}\vert ^{\beta }\nabla v) = 0\) behaves on concentric balls. When these balls are centred on \(\partial {\mathbb {R}}^{m+1}_+\), we have the following.
Lemma 2.8
Let \(B_R(x_0)\subset {\mathbb {R}}^{m+1}\) with \( x_0 \in \partial {\mathbb {R}}^{m+1}_+ \) and \(R\le 1\). Suppose \(v \in W^{1,2}_{\beta }(B_R(x_0);{\mathbb {R}}^n)\) is a weak solution of \(\text {div}(\vert x_{m+1}\vert ^{\beta }\nabla v) = 0\) in \(B_R(x_0)\). If \(\beta \in (-1,0]\), or if \(\beta \in (0,1)\) and v is symmetric with respect to \(\partial {\mathbb {R}}^{m+1}_+\) in \(B_R(x_0)\), then
for every \(0<s\le r\le R\).
Remark 2.9
It appears that Lemma 2.8 is a particular case of Theorem 2.6 of [3]. The function
fulfils the hypothesis of Lemma 2.8, with the exception that it is not symmetric with respect to \(\partial {\mathbb {R}}^{m+1}_+\) when \(\beta \in (0,1)\). Furthermore, \(s^{-(1+m+\beta )}\int _{B_s(0)}\vert x_{m+1}\vert ^{\beta }\vert \nabla {\tilde{v}}\vert ^2\mathrm {d}x = C(m,\beta )s^{-2\beta }\). It follows that \({\tilde{v}}\) does not satisfy the conclusion of the lemma (e.g on \(B_1(0)\)) when \(\beta \in (0,1)\). Hence Lemma 2.8 does not hold in general when \(\beta \in (0,1)\). This also potentially affects the aforementioned theorem.
The remainder of this section is predominantly devoted to a proof of Lemma 2.8. We also establish a similar monotonicity formula for balls \(B_R(x_0)\) with \((x_0)_{m+1} \ge \theta R\) for \(\theta \ge 2\) and give explicit dependence on \(\theta \). Our method is essentially that used to show the well known monotonicity results for the average energy of sub-harmonic functions, which is based on the observation \({\varDelta }\vert \nabla v\vert ^2 \ge 0\) if \({\varDelta }v = 0\). When \(\beta = 0\) our result reduces to the usual growth formula for the average energy of sub-harmonic functions.
Define \(v^{*}:= \vert x_{m+1}\vert ^{\beta }\partial _{m+1} v\). This function will be integral to our proof of Lemma 2.8 because, as we will see in more detail later, it satisfies \(\text {div}(\vert x_{m+1}\vert ^{-\beta }\nabla v^{*}) = 0\) when \(\text {div}(\vert x_{m+1}\vert ^{ \beta }\nabla v) = 0\). The fact that if \(\text {div}(x_{m+1}^{\beta }\nabla v)= 0 \) in \({\mathbb {R}}^{m+1}_+\) then \(\text {div}(x_{m+1}^{-\beta }\nabla v^{*})= 0 \) was first observed by Caffarelli and Silvestre [2].
First, we consider the regularity of solutions of \(\text {div}(\vert x_{m+1}\vert ^{\beta }\nabla v) = 0\); the following regularity results are known and have essentially been obtained in, for example [4]. We give a proof regardless, as an illustration that the method of difference quotients works essentially unchanged in the directions \(x_1,\ldots ,x_m\), which we will take advantage of when considering the regularity of manifold valued minimisers of \(E^{\beta }\).
Let \(i = 1,\ldots ,m\), \({\varOmega }\subset {\mathbb {R}}^{m+1}\) and let \(h \in {\mathbb {R}}{\backslash }\{0\}\). Define the difference quotient of \(v{:}\,{\varOmega }\rightarrow {\mathbb {R}}^n\) by \({\varDelta }_i^h v(x) = h^{-1}(v(x+he_i)-v(x))\) where \(e_i\) denotes the ith basis vector in \({\mathbb {R}}^{m+1}\) and \(\text {dist}(x,\partial {\varOmega })<\vert h\vert \).
Lemma 2.10
Let \({\varOmega }\subset {\mathbb {R}}^{m+1}\) be open and \(v \in {W}_{\beta }^{1,2}({\varOmega };{\mathbb {R}}^n)\). Then for any \(i = 1,\ldots ,m\) we have \({\varDelta }_i^{h}v \in L_{\beta }^2(K;{\mathbb {R}}^n)\) for any compact \(K \subset {\varOmega }\), provided \(\vert h \vert < \text {dist}(K, \partial {\varOmega })\). In particular,
Proof
This proof follows the proof of Lemma 7.23 in [15]. We assume that \(h \ge 0\), the argument for negative h is analogous. Let \(v \in C^{1}({\varOmega };{\mathbb {R}}^n)\cap {W}_{\beta }^{1,2}({\varOmega };{\mathbb {R}}^n)\). Using the notation \(K_h = \{ x \in {\mathbb {R}}^{m+1}{:}\,\text {dist}(x,K)\ \le h\} \) and noting \(K_h \subset {\varOmega }\), by Fubini’s Theorem and the compactness of K, for h with \(\vert h\vert <\text {dist}(K, \partial {\varOmega })\) and any \(i = 1,\ldots ,m\), we calculate
We deduce the result for \(v \in {W}_{\beta }^{1,2}({\varOmega };{\mathbb {R}}^n)\) by approximation. \(\square \)
Next we prove a criterion for the existence of weak derivatives in the directions \(x_i\) for \(i = 1,\ldots ,m\).
Lemma 2.11
Suppose \({\varOmega }\subset {\mathbb {R}}^{m+1} \) is open and bounded and let \(v \in L_{\beta }^2({\varOmega };{\mathbb {R}}^n)\). For any \(i = 1,\ldots ,m\), suppose there exist constants \(M>0\) and \({\tilde{h}}>0\) such that
for every \(h\ne 0\) with \(\vert h\vert <{\tilde{h}}\) and compact \(K \subset {\varOmega }\) with \(\mathrm {dist}(K,\partial {\varOmega })>\vert h\vert \). Then the weak derivative \(\partial _iv\) exists in \({\varOmega }\) and satisfies
Proof
First choose a sequence \((h_k)_{k\in {\mathbb {N}}}\) with \(h_k\rightarrow 0\). We discard \(h_k\) with \(\vert h_k\vert \ge {\tilde{h}} \) and re-index to \(k \in {\mathbb {N}}\). Define \(v_i^{h_k}(x) = {\varDelta }_i^{h_k}v(x)\) when \(x \in {\varOmega }\) and \(\text {dist}(x,\partial {\varOmega })\ge 2\vert h_k\vert \) and \(v_i^{h_k} = 0\) otherwise. It follows that \(\{v_i^{h_k}\}_{k \in {\mathbb {N}}}\) is a bounded sequence in \(L^2_{\beta }({\varOmega };{\mathbb {R}}^{n})\). Hence there is a subsequence, which we index again by \(k \in {\mathbb {N}}\), such that \(h_k\rightarrow 0\) and \(v_i^{h_k} \rightarrow {\tilde{v}}_i\) weakly in \(L^2_{\beta }({\varOmega };{\mathbb {R}}^{n})\). Furthermore, this convergence, together with the weak lower semi-continuity of a Hilbert space norm, guarantees that \(\int _{{\varOmega }}x_{m+1}^{\beta }\left| {\tilde{v}}_i\right| ^2\mathrm {d}x \le M\). Note that Hölder’s inequality implies \(L^2_{\beta }({\varOmega };{\mathbb {R}}^n) \subset L^p ({\varOmega };{\mathbb {R}}^n) \) for some \(p \in (1,2]\) depending on \(\beta \). Thus each linear functional on \( L^p ({\varOmega };{\mathbb {R}}^n) \) restricts to a linear functional on \(L^2_{\beta }({\varOmega };{\mathbb {R}}^n)\) and \(v_i^{h_k}\) converges to \({\tilde{v}}_i\) weakly in \(L^p ({\varOmega };{\mathbb {R}}^n) \). As in the proof of Lemma 7.24 in [15], it follows that \({\tilde{v}}_i\) is the weak derivative \(\partial _i v\). \(\square \)
We now establish the regularity properties of solutions of \(\text {div}(\vert x_{m+1}\vert ^{\beta }\nabla v) = 0\) needed for the proof of Lemma 2.8.
Lemma 2.12
Let \(v \in W^{1,2}_{\beta }(B_R(x_0);{\mathbb {R}}^n)\) and suppose v is a weak solution of \(\text {div}(\vert x_{m+1}\vert ^{\beta }\nabla v) = 0\) in \(B_R(x_0)\). For every \(r<R\) and \(i = 1,\ldots ,m\) it follows that \(\partial _iv \in W^{1,2}_{\beta }(B_r(x_0);{\mathbb {R}}^n)\), \(\partial _i v \) is a weak solution of \(\text {div}(\vert x_{m+1}\vert ^{\beta }\nabla v) = 0\) in \(B_r(x_0)\) and \(\partial _iv\) is locally Hölder continuous in \(B_R(x_0)\). In addition, \(v^{*} \in W^{1,2}_{-\beta }(B_r(x_0);{\mathbb {R}}^n)\) and \(v^{*}\) is a weak solution of \(\text {div}(\vert x_{m+1}\vert ^{-\beta }\nabla v^{*}) = 0\) in \(B_r(x_0)\) and \(v^{*}\) is locally Hölder continuous in \(B_R(x_0)\).
Proof
Elliptic regularity theory shows that v is smooth in \(B_R(x_0){\backslash } \partial {\mathbb {R}}^{m+1}_+\) [15]. Observe that v satisfies
for every \(\phi \in W^{1,2}_{\beta ,0}(B_R(x_0);{\mathbb {R}}^n)\) by approximation. Let \(r<R\) and choose \(\eta \in C_0^{\infty }(B_R(x_0))\) with \(\eta \equiv 1\) in \(B_r(x_0)\), \(\eta \equiv 0\) in \( B_R(x_0){\backslash } B_{r + \frac{R-r}{2}}(x_0)\), \(0 \le \eta \le 1\) and \(\vert \nabla \eta \vert \le \frac{C}{R-r}\). Let \({\varDelta }_i^hv\) be the difference quotient of v for some \(i = 1,\ldots ,m\) and suppose \(\vert h\vert < \frac{R-r}{4}\). Then \(\phi = -{\varDelta }_i^{-h}(\eta ^2{\varDelta }_i^hv) \in W^{1,2}_{\beta ,0}(B_R(x_0);{\mathbb {R}}^n)\) is an admissible test function for (2.17) and an application of Young’s inequality, \(ab \le \delta \frac{a^2}{2} + \delta ^{-1}\frac{b^2}{2} \) for \(a,b\ge 0\) and \(\delta >0\), together with an integration by parts and Lemma 2.10 implies
Since \(\eta \equiv 1\) in \(B_r(x_0)\), choosing \(\delta = \frac{R-r}{2C}\) we deduce that
The right hand side above is independent of h and thus Lemma 2.11 implies the weak derivative \(\nabla \partial _iv\) exists and is in \(L^2_{\beta }(B_r(x_0);{\mathbb {R}}^{(m+1)n})\). Hence \(\partial _iv \in W^{1,2}_{\beta }(B_r(x_0);{\mathbb {R}}^n)\) for every \(r<R\). We integrate by parts in (2.17) to see that \(\partial _iv\) is a weak solution of \(\text {div}(\vert x_{m+1}\vert ^{\beta }\nabla v) = 0\) in \(B_r(x_0)\) for every \(r<R\). It follows from [13, Theorem 2.3.12] that each \(\partial _iv\) is locally Hölder continuous in \(B_R(x_0)\). We inductively deduce that for any multi-index \(\alpha '\) with \(\alpha '_{m+1} = 0\), \(D^{\alpha '}v \in W^{1,2}_{\beta }(B_r(x_0);{\mathbb {R}}^n)\) is a weak solution of \(\text {div}(\vert x_{m+1}\vert ^{\beta }\nabla D^{\alpha '}v) = 0\) in \(B_r(x_0)\) for every \(r<R\) and \(D^{\alpha '}v\) is locally Hölder continuous in \(B_R(x_0)\).
Since \(\partial _i v \in W^{1,2}_{\beta }(B_r(x_0);{\mathbb {R}}^n)\) for every \(r<R\) and \(i = 1,\ldots ,m\), we have
We also have \({\varDelta }' v \in W^{1,2}_{\beta }(B_r(x_0);{\mathbb {R}}^n)\), where \({\varDelta }'\) is the Laplace operator with respect to the variables \(x_1,\ldots ,x_m\). Furthermore, as v solves \(\text {div}(\vert x_{m+1}\vert ^{\beta }\nabla v) = 0\) classically in \(B_R(x_0){\backslash } \partial {\mathbb {R}}^{m+1}_+\), we have \(\vert x_{m+1}\vert ^{-\beta } \partial _{m+1}v^{*} = -{\varDelta }' v \in W^{1,2}_{\beta }(B_r(x_0) ;{\mathbb {R}}^n)\) for every \(r<R\). Hence
Together, (2.18) and (2.19) imply \(\int _{B_r(x_0)}\vert x_{m+1}\vert ^{-\beta }\vert \nabla v^{*} \vert ^2\mathrm {d}x<\infty \) for every \(r<R\). Moreover, we have
since \(v \in W^{1,2}_{\beta }(B_R(x_0);{\mathbb {R}}^n)\). We can directly verify that \(\partial _iv^*\), \(i = 1,\ldots ,m\), are the weak derivatives of \(v^*\) and omit the details. Now consider \(\partial _{m+1}v^*\). Let \(\nabla '\) denote the gradient operator with respect to the variables \(x_1,\ldots ,x_m\) and let \(\psi \in C_0^{\infty }(B_R(x_0);{\mathbb {R}}^n)\). Since \(\vert x_{m+1}\vert ^{-\beta } \partial _{m+1}v^{*} = -{\varDelta }' v \in W^{1,2}_{\beta }(B_r(x_0) ;{\mathbb {R}}^n)\) and v is a weak solution of \(\text {div}(\vert x_{m+1}\vert ^{\beta }\nabla v)=0\), we see that
It follows that \(v^* \in W^{1,2}_{-\beta }(B_r(x_0);{\mathbb {R}}^n)\) for every \(r<R\). In a similar manner we calculate
Hence \(v^{*}\) is a weak solution of \(\text {div}(\vert x_{m+1}\vert ^{\beta }\nabla v^{*}) = 0\) in \(B_r(x_0)\) for every \(r<R\). It follows from [13, Theorem 2.3.12] that \(v^{*}\) is locally Hölder continuous in \(B_R(x_0)\). \(\square \)
Corollary 2.13
Suppose \(v \in W^{1,2}_{\beta }(B_R(x_0);{\mathbb {R}}^n)\) and assume v weakly satisfies \(\text {div}(\vert x_{m+1}\vert ^{\beta }\nabla v) = 0\) in \(B_R(x_0)\). Then the derivatives \(D^{\alpha '}v\), where \(\alpha '\) is a multi-index with \((\alpha ')_{m+1} = 0\), are elements of \(W^{1,2}_{\beta }(B_r(x_0);{\mathbb {R}}^n)\) and weak solutions of \(\text {div}(\vert x_{m+1}\vert ^{\beta }\nabla v) = 0\) in \(B_r(x_0)\) for every \(r<R\) and are locally Hölder continuous in \(B_R(x_0)\). Furthermore, the functions \((D^{\alpha '}v)^{*}:= \vert x_{m+1}\vert ^{\beta }\partial _{m+1}D^{\alpha '}v\) are elements of \(W^{1,2}_{-\beta }(B_r(x_0);{\mathbb {R}}^n)\) and weak solutions of \(\text {div}(\vert x_{m+1}\vert ^{-\beta }\nabla (D^{\alpha '}v)^{*}) = 0\) in \(B_r(x_0)\) for every \(r<R\) and are locally Hölder continuous in \(B_R(x_0)\).
Proof
This follows from a direct application of Lemma 2.12. \(\square \)
Next we record a condition on the integral of the normal derivative of Sobolev functions which implies the type of monotonicity we want to establish in Lemma 2.8.
Lemma 2.14
Suppose that \(v \in W^{1,2}_{\beta }(B_R(x_0))\), where \(x_0 \in \partial {\mathbb {R}}^{m+1}_+\), and that
for almost every \(\rho \in (0,R)\), where \(\nu \) is the outward pointing unit normal on \(\partial B_{\rho }(x_0)\), then
for every \(0<s \le r \le R\).
Proof
This proof follows the proof of Theorem 2.1 in Section 2 of [15] and the proof of Proposition 2.2 in Section III of [14]. For almost every \(0< s\le r <R\), using Fubini’s Theorem, we see that
Define the absolutely continuous function \(f(r) = \int _{B_r(x_0)}\vert x_{m+1}\vert ^{\beta }v\mathrm {d}x\) for \(0\le r\le R\). Using (2.22) we calculate
for \(0<r< R\). It follows that \((r^{-(1+m+\beta )} f(r))' \ge 0\) and integrating between \(s\le r\le R\) completes the proof. \(\square \)
Proof of Lemma 2.8
Note that v is smooth in \(B_R(x_0){\backslash } \partial {\mathbb {R}}^{m+1}_+\). Hence \(\text {div}(\vert x_{m+1}\vert ^{ \beta }\nabla \partial _iv) = 0\) and \(\text {div}(\vert x_{m+1}\vert ^{-\beta }\nabla v^{*}) = 0\) classically in this set. Furthermore, we have
so that \(\text {div}(\vert x_{m+1}\vert ^{ \beta }\nabla \partial _{m+1} v ) =\beta \vert x_{m+1}\vert ^{ \beta - 2}\partial _{m+1} v\) in \(B_R(x_0){\backslash } \partial {\mathbb {R}}^{m+1}_+\). Hence on \(B_R(x_0){\backslash } \partial {\mathbb {R}}^{m+1}_+\) we have
and, when \(\beta \in (0,1)\),
classically where \(i = 1,\ldots ,m\).
Fix \(R>r>\varepsilon >0\) and let \(B_r^{\varepsilon }(x_0) = B_r(x_0) \cap \{ x \in {\mathbb {R}}^{m+1} {:}\,\vert x_{m+1}\vert \ge \varepsilon \}\). Using the divergence theorem, we calculate
and, when \(\beta \in (0,1)\),
where \({\mathbb {1}}_{\overline{B_r^{\varepsilon }(x_0)}}\) is the indicator function of \(\overline{B_r^{\varepsilon }(x_0)}\), \(\nu \) is the outward unit normal on \(\partial B_r(x_0)\), \(\mathrm {d}S\) is the Lebesgue measure on \(\partial B_r(x_0)\) and \(B^m_{s}(x_0)= \{x \in {\mathbb {R}}^m{:}\,\vert x - x_0\vert <s\}\).
We consider the terms on the right hand side of (2.23)–(2.25) separately with a view to taking the limit as \(\varepsilon \rightarrow 0^+\). Lemma 2.12 and Corollary 2.13 imply that \( \partial _i v, \partial _j\partial _iv \in W^{1,2}_{\beta }(B_r(x_0);{\mathbb {R}}^n)\) and are locally Hölder continuous in \(B_R(x_0)\) for \(i,j = 1,\ldots ,m\). The lemma and corollary further imply that \(v^{*}, (\partial _iv)^{*} \in W^{1,2}_{-\beta }(B_r(x_0);{\mathbb {R}}^n)\), where \(i = 1,\ldots ,m\), are locally Hölder continuous in \(B_R(x_0)\) and hence uniformly continuous in \(\overline{B_r(x_0)}\). We can therefore check that \(\vert \partial _iv\vert ^2 \in W^{1,2}_{\beta }(B_r(x_0))\), integrating over \(B^{\varepsilon }_r(x_0)\) and letting \(\varepsilon \rightarrow 0^+\) for the \(m+1\)th derivative. It follows that
where C is a positive constant that may depend on r but is independent of \(\varepsilon \). Furthermore, since \(\partial _iv\) and \(( \partial _i v)^{*}\) are uniformly continuous in \(\overline{B_r(x_0)}\), we see that
for \( (x',0) \in \overline{B_r(x_0)}\).
Since each \( \partial _j\partial _iv \in W^{1,2}_{\beta }(B_r(x_0);{\mathbb {R}}^n)\) is locally Hölder continuous in \(B_R(x_0)\) for \(i,j=1,\ldots ,m\) and \(\text {div}(\vert x_{m+1}\vert ^{\beta }\nabla v)= 0\) classically in \(B_R(x_0){\backslash }\partial {\mathbb {R}}^{m+1}_+\), we have \(\vert x_{m+1}\vert ^{-\beta }\partial _{m+1} v^{*} = - {\varDelta }' v \in W^{1,2}_{\beta }(B_r(x_0);{\mathbb {R}}^n)\) is uniformly continuous in \(\overline{B_r (x_0)}\) for every \(r<R\). We can hence check \(\vert v^{*}\vert ^2 \in W^{1,2}_{-\beta }(B_r(x_0))\). We also have
and
for \( (x',0) \in \overline{B_r(x_0)}\).
In order to derive similar conclusions to (2.26) and (2.27) for the constituent integrands of (2.25), we assume that \(\beta \in (0,1)\) and that v is symmetric with respect to \(\partial {\mathbb {R}}^{m+1}_+\), namely \(v(x',x_{m+1}) =v(x',-x_{m+1}) \) for every \((x',x_{m+1}) \in B_R(x_0)\). The symmetry of v implies \(v^{*}\) must be odd with respect to \(\partial {\mathbb {R}}^{m+1}_+\), that is \(v^{*}(x',x_{m+1}) =-v^{*}(x',-x_{m+1}) \) for every \((x',x_{m+1}) \in B_R(x_0)\) and hence, as it is also continuous in \(B_R(x_0)\) we have \(v^{*}(x',0)=0\) for every \((x',0) \in B_R(x_0)\).
Fix \((x',0) \in B_R(x_0)\) and note that \((x',0) \in B_r(x_0)\) for some \(r<R\) and choose h with \(\vert h\vert \) sufficiently small as to ensure \((x',h) \in B_r(x_0)\). We see that
where \(x_{m+1}\) with \(\vert x_{m+1}\vert \in (0,\vert h\vert )\) and \(\xi \) with \(\vert \xi \vert \in (0,\vert x_{m+1}\vert )\) are chosen such that the Mean Value Theorem holds. Thus we see that \(\partial _{m+1} v(x',0) = 0 \) classically for \((x',0) \in B_R(x_0)\). Analogous calculations to those on the right hand side above show that \( \partial _{m+1} v\) is continuous at \((x',0)\) and hence continuous in \(B_R(x_0)\). We also have
for \((x',x_{m+1}) \in \overline{B_r(x_0)}{\backslash }\partial {\mathbb {R}}^{m+1}_+\) and hence
in \(\overline{B_r(x_0)}{\backslash }\partial {\mathbb {R}}^{m+1}_+\), where \(\xi \) is chosen with \(\vert \xi \vert \in (0,\vert x_{m+1}\vert )\) such that the Mean Value Theorem holds. It follows that \(\vert x_{m+1}\vert ^{\beta }\partial ^2_{m+1} v\) is essentially bounded in \(\overline{B_r(x_0)}\) and \(\partial B_r(x_0)\).
The preceding discussion implies \(\vert \partial _{m+1}v\vert ^2 \in W^{1,2}_{\beta }(B_r(x_0))\) and
on \(\overline{B_r(x_0)}{\backslash }\partial {\mathbb {R}}^{m+1}_+\). Furthermore, using the symmetry of v, we see that
for \( (x',0) \in \overline{B_r(x_0)}\).
Using Lebesgue’s Dominated Convergence Theorem, we combine (2.23) with (2.26) and (2.27), (2.24) with (2.28) and (2.29) and (2.25) with (2.30) and (2.31) to see that
and, when \(\beta \in (0,1)\) and v is symmetric with respect to \(\partial {\mathbb {R}}^{m+1}_+\),
respectively. Noting that \(\vert x_{m+1}\vert ^{-\beta }\vert v^{*}\vert ^2 = \vert x_{m+1}\vert ^{\beta }\vert \partial _{m+1}v \vert ^2\), we apply Lemma 2.14, to see that
and, when \(\beta \in (0,1)\) and v is symmetric with respect to \(\partial {\mathbb {R}}^{m+1}_+\),
for every \(0<s\le r< R\). We apply Lebesgue’s Dominated Convergence Theorem, sending \(r \rightarrow R^{- }\) and combine (2.32), (2.33) and (2.34) to see the Lemma holds for \(0<s\le r\le R\). \(\square \)
Remark 2.15
We observe from (2.33) and (2.34) that without the symmetry condition on v, the monotonicity of the average energy of \(\vert \partial _{m+1}v\vert ^2\) is better than required if \(\beta \in (-1,0]\) and may be worse than required if \(\beta \in (0,1)\); the symmetry of v resolves this issue by implying continuity of \(\partial _{m+1}v\) on \(\partial {\mathbb {R}}^{m+1}_+\) which, as the function given in Remark 2.9 illustrates, cannot be expected in general.
2.5 Interior monotonicity formula
We need a counterpart to Lemma 2.8 for balls in the interior of \({\mathbb {R}}^{m+1}_+\). Since the weight \(x_{m+1}^{\beta }\) is not scale invariant on a (Euclidean) ball \(B_{\rho }(y)\) with \(y_{m+1}\ge 2\rho \), we only expect monotonicity of the average energy (with respect to the Lebesgue measure) of solutions to \(\text {div}(x_{m+1}^{\beta }\nabla v) = 0\) up to a correcting factor. As the radius tends to zero, the coefficients of uniform ellipticity for the preceding equation tend to constants. Accordingly, the correcting factor becomes smaller with the radius. General monotonicity-type formulas for linear uniformly elliptic equations are available in [14] for example. We perform the following calculations in order to determine how the correcting factor behaves explicitly as the radius decays geometrically. We first establish monotonicity on the boundary of concentric balls.
Lemma 2.16
Let \(\beta \in (-1,1)\), \(B_{R}(y) \subset {\mathbb {R}}^{m+1}_+\) with \(\overline{B_{R}(y)}\subset {\mathbb {R}}^{m+1}_+\) and suppose \(v \in C^2(B_R(y);{\mathbb {R}}^n) \) satisfies \(\text {div} (x_{m+1}^{\beta }\nabla \vert v\vert ^2) \ge 0\) classically in \(B_{R}(y) \). Then, letting \(\mathrm {d}S\) denote Lebesgue surface measure and \(\text {sgn}\) denote the sign function, for \(0< s \le r < R\) we have
Proof
Let \(\rho < R\). We calculate \(0 \le \int _{\partial B_{\rho }(y)}x_{m+1}^{\beta } \nu \cdot \nabla \vert v \vert ^2 \mathrm {d}S(x)\), where \(\nu \) is the unit outward normal on \(\partial B_{\rho }(y)\). Now using variables \(\rho = \vert x-y\vert \) and \(\omega = \frac{x-y}{\rho }\) we have
We define \(f(\rho ) =\rho ^{-m}\int _{\partial B_{\rho }(y)} x_{m+1}^{\beta } \vert v \vert ^2 \mathrm {d}S(x)\) and divide by \(\rho ^{m}\) to see that
Hence \(0 \le \left( (y_{m+1} - \text {sgn}(\beta )\rho )^{-\beta } f(\rho )\right) '\) and integrating between \(s<r\) concludes the proof. \(\square \)
With this lemma in hand we can establish the following counterpart to Lemma 2.8.
Lemma 2.17
Let \(B_R(y) \subset {\mathbb {R}}^{m+1}\) with \(y_{m+1} \ge \theta R\) for \(\theta \ge 2\). Suppose \(v \in C^2(B_R(y);{\mathbb {R}}^n)\cap W^{1,2}_{\beta }(B_R(y);{\mathbb {R}}^n) \) satisfies \(\text {div} (x_{m+1}^{\beta }\nabla v) = 0\) in \(B_R(y)\). Then there exists \(C = C(m)\) such that
Proof
Let \(g(r):= \int _{B_r(y)}x_{m+1}^{\beta }\vert f\vert ^2\mathrm {d}x\) where f satisfies the assumptions of Lemma 2.16 and \(r \in (0,R)\). Then we have
Hence
where \(s \in (0,r)\) is such that the Mean Value Theorem holds for the function \(r \mapsto (1-\text {sgn}(\beta )\frac{r}{y_{m+1}})^{\beta }\). Now recall that \(y_{m+1}\ge \theta R\ge \theta r\) for \(\theta \ge 2\). We hence find
It follows that \(0 \le (r^{-(m+1)}r^{ \frac{\vert \beta \vert (m+1)}{\theta - 1} }g (r))'\) and consequently, if \( \frac{R}{2}\le r <R \), we have
where \(\xi \in (1,2)\) is such that the Mean Value theorem holds for the function \(t\mapsto t^{ \frac{\vert \beta \vert (m+1)}{\theta - 1} }\). Since \(\text {div}(x_{m+1}^{\beta } \nabla v) = 0\), as observed in the proof of Lemma 2.8, we have \(\text {div}(x_{m+1}^{\beta } \nabla \vert \partial _iv\vert ^2) \ge 0\) for \(i = 1,\ldots ,m\) and \(\text {div}(x_{m+1}^{-\beta } \nabla \vert x_{m+1}^{\beta }\partial _{m+1}v\vert ^2) \ge 0\). We apply (2.36) with \(f = \partial _iv\) for \(i = 1,\ldots , m\), and with \(f = x_{m+1}^{\beta }\partial _{m+1}v\) and \(-\beta \) in place of \(\beta \) and combine the results, letting \(r \rightarrow R^{-}\) to conclude the proof. \(\square \)
2.6 Solutions of the linear degenerate Dirichlet problem
We require further results regarding the following Dirichlet problem: solve
for a given \(\phi \), where \(x_0 \in {\mathbb {R}}^{m+1}\), in order to apply a version of the method of harmonic replacement in the proof of Lemma 4.23. A weak solution of (2.37) is a \(v \in W_{\beta }^{1,2}(B_R(x_0);{\mathbb {R}}^n)\) which weakly satisfies \(\text {div}(\vert x_{m+1}\vert ^{\beta }\nabla v) = 0\) in \(B_R(x_0) \) with \(v - \phi \in W_{\beta ,0}^{1,2}(B_R(x_0);{\mathbb {R}}^n)\). We collect the results we require, which can be found in [19], in the form of a lemma.
Lemma 2.18
Suppose \( \phi \in W^{1,2}_{\beta }(B_R(x_0);{\mathbb {R}}^n)\). Then there exists a \(v\in W^{1,2}_{\beta }(B_R(x_0);{\mathbb {R}}^n)\) which is a weak solution of the Dirichlet problem (2.37). Any such solution is unique and continuous in \(B_R(x_0)\), if \(\phi \in C(\overline{B_R(x_0)};{\mathbb {R}}^n)\) then \(v(x) \rightarrow \phi (z)\) as \(x \rightarrow z\) for \(z \in \partial B_R(x_0)\) and the weak maximum principle
and weak minimum principle
both hold, where we take the maximum and minimum component-wise. If \(w \in W^{1,2}_{\beta }(B_R(x_0);{\mathbb {R}}^n)\) also satisfies \(w - \phi \in W^{1,2}_{\beta ,0}(B_R(x_0);{\mathbb {R}}^n)\) then
Proof
Since \(\vert x_{m+1}\vert ^{\beta }\) is of Muckenhoupt class \(A_2\) it follows from 1.6 of [19] that \(\vert x_{m+1}\vert ^{\beta }\) is a 2-admissible weight so we may apply the theory of [19]. Aside from the minimising property of v, the assertions of the lemma are consequences of Theorem 3.70, Corollary 6.32, the strong maximum principle 6.5 and lastly 3.17 in [19]. If v is a weak solution of (2.37) for a given \(\phi \) and \(w \in W_{\beta }^{1,2}(B_R(x_0);{\mathbb {R}}^n)\) with \(w-\phi \in W_{\beta ,0}^{1,2}(B_R(x_0);{\mathbb {R}}^n)\) then \(w-v \in W_{\beta ,0}^{1,2}(B_R(x_0);{\mathbb {R}}^n)\). Hence, by approximation, we have
so that
which concludes the proof. \(\square \)
The uniqueness of solutions to the Dirichlet problem (2.37) implies that solutions with boundary data which are symmetric with respect to \(\partial {\mathbb {R}}^{m+1}_+\) are themselves symmetric. More precisely, we have the following.
Lemma 2.19
Let \(x_0 \in \partial {\mathbb {R}}^{m+1}_+\) and suppose \(v, \phi \in W^{1,2}_{\beta }(B_R(x_0);{\mathbb {R}}^n)\) and v is a weak solution of the Dirichlet problem (2.37) with \(\phi \) as boundary data. Let \(\phi \in C(\overline{B_R(x_0)};{\mathbb {R}}^n)\) and suppose \(\phi (x',x_{m+1}) = \phi (x',-x_{m+1})\) for every \((x',x_{m+1}) \in \overline{B_R(x_0)}\). Then \(v(x',x_{m+1})= v(x',-x_{m+1})\) for every \((x',x_{m+1}) \in B_R(x_0)\).
Proof
The continuity of \(\phi \) in \(\overline{B_R(x_0)}\), combined with an application of Lemma 2.18, implies that v and, consequently, \({\tilde{v}}(x',x_{m+1}):= v(x',-x_{m+1})\) are continuous in \(\overline{B_R(x_0)}\). We observe that \({\tilde{v}} \in W^{1,2}_{\beta }(B_R(x_0);{\mathbb {R}}^{n})\) weakly satisfies \(\text {div}(\vert x_{m+1}\vert ^{\beta }\nabla {\tilde{v}}) = 0\) in \(B_R(x_0)\) and \({\tilde{v}}\vert _{\partial B_R(x_0)} = \phi \vert _{\partial B_R(x_0)}\) so that \({\tilde{v}} - \phi \in W^{1,2}_{\beta ,0}(B_R(x_0);{\mathbb {R}}^n)\). Hence v and \({\tilde{v}}\) solve the same Dirichlet problem; solutions to this problem are unique by Lemma 2.18 and thus \({\tilde{v}} = v\). \(\square \)
3 Intrinsic fractional harmonic maps
We assume, translating N if necessary, that \(0 \in N\). For reasons discussed in the introduction we always assume \(m \ge 2\); when \(m \ge 3\) we let \(\beta \in (-1,1)\) and when \(m = 2\) we let \(\beta \in (-3^{-1},1)\). This allows us to apply Lemma 2.3 and we can define
Henceforth, in this section and Sect. 4, we fix an open \({\mathcal {O}} \subset \partial {\mathbb {R}}^{m+1}_+\) such that a continuous linear trace operator with respect to \({\mathcal {O}}\) exists; we do not assume \({\mathcal {O}}\) is necessarily the whole of \(\partial {\mathbb {R}}^{m+1}_+\). We can, for example, obtain such a trace operator \(T{:}\,{\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;{\mathbb {R}}^n)\rightarrow L^p({\mathcal {O}};{\mathbb {R}}^n)\), where \(p = p(\beta ) \in (1,2]\), by combining Lemmata 2.1 and 2.3 with [12, Section 4.3, Theorem 1] whenever \({\mathcal {O}}\) is contained in the boundary of a Lipschitz \({\varOmega }\subset {\mathbb {R}}^{m+1}_+\). Define
for \(u \in T({\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N))\). Recall from the introduction that \(I^{\beta }\) serves as an intrinsic energy for u; it does not depend on the choice of embedding of N into Euclidean space. Moreover, \(I^{\beta }\) coincides with the square of the fractional Sobolev norm \(\vert \vert u\vert \vert _{{\dot{H}}^{ \frac{1-\beta }{2}}(\partial {\mathbb {R}}^{m+1}_+ ;{\mathbb {R}}^n)}\) when \(N = {\mathbb {R}}^n\) and \({\mathcal {O}} = \partial {\mathbb {R}}^{m+1}_+\).
For every \(u \in T({\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N))\), an application of the direct method of the calculus of variations shows that there exists \(v \in {\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N)\) with \(Tv = u\), where T is the trace operator with respect to \({\mathcal {O}}\), such that \(I^{\beta }(u) = E^{\beta }(v)\). For a given u, such a v is referred to henceforth as a minimal harmonic map. Any minimal harmonic map v is weakly harmonic in \({\mathbb {R}}^{m+1}_+\) with respect to the metric represented in Euclidean coordinates by \(x_{m+1}^{\alpha }\delta _{ij}\), where \(\beta = \frac{\alpha (m-1)}{2}\); the Dirichlet energy on \({\mathbb {R}}^{m+1}_+\) for this metric is precisely \(E^{\beta }\). Such a v therefore satisfies
for every \(\phi \in C^{\infty }_0({\mathbb {R}}^{m+1}_+;{\mathbb {R}}^n)\), where A is the second fundamental form of N and
Formally, if v is sufficiently regular in \({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}\), we calculate
for every \(\phi \in C_0^{\infty }({\mathcal {O}};{\mathbb {R}}^n)\) and any \(\psi \in {\mathcal {D}}_+({\mathbb {R}}^{m+1}_+;{\mathbb {R}}^n)\) with \(\psi (x',0) = \phi (x')\), where \(\mathrm {d}x'\) is the Lebsegue measure on \({\mathbb {R}}^m\). In general, the integral in (3.1) defines a distribution on \({\mathcal {O}}\) given by
for \(\phi \in C^{\infty }_0({\mathcal {O}};{\mathbb {R}}^n)\). This observation allows us, analogously to [30, Proposition 1.1], to identify a superdifferential for \(I^{\beta }\). Recall that since N is compact, Theorem 1 in Section 2.12.3 of [39] gives a tubular neighbourhood of N, which has the form \(U_{\delta }(N) = \{x \in {\mathbb {R}}^n{:}\,\text {dist}(x,N)<\delta \}\) for a \(\delta = \delta (N)>0\), and a smooth map \(\pi _N:U_{\delta }(N)\rightarrow N\) such that \(\vert \pi _N(y) -y\vert = \text {dist}(y,N)\) for every \(y \in U_{\delta }(N)\). Using the same method of proof as [30, Proposition 1.1] we deduce the following.
Lemma 3.1
Let \(u \in T({\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N))\) and \(v \in {\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N)\) be a minimal harmonic map with \(Tv = u\). Then for \(\phi \in C_{0}^{\infty }({\mathcal {O}};{\mathbb {R}}^n)\),
as \(t \rightarrow 0\).
It follows from this proposition that if \(\left. \frac{\partial }{\partial t}\right| _{t=0}I^{\beta }(\pi _N(u + t\phi ))\) exists then it is equal to \(-\partial _{m+1}^{\beta }v(\phi )\) where v is any minimal harmonic map with \(Tv = u\); this indicates a candidate for the first variation of \(I^{\beta }\).
Definition 3.2
Let \(\beta \in (-1,1)\) and \({\mathcal {D}}_{\beta }\) be the collection of all \(u \in T({\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N))\) such that there exists a distribution \(\lambda _{\beta } \in \left( C_0^{\infty }({\mathcal {O}};{\mathbb {R}}^n)\right) ^*\) with \(\lambda _{\beta } = -\partial _{m+1}^{\beta }v\) for every minimal harmonic map \(v \in {\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N)\) with \(Tv = u\). Then we may define a map \({\varLambda }_{\beta }{:}\,{\mathcal {D}}_{\beta } \rightarrow \left( C_0^{\infty }({\mathcal {O}};{\mathbb {R}}^n)\right) ^*{:}\,u \mapsto \lambda _{\beta } = {\varLambda }_{\beta }u\).
In [30, Theorem 1.1] Moser showed that \({\varLambda }_{0}\) is the first variation of \(I^0\). The method of proof of Moser’s theorem, applied with the Lebesgue measure \(\mathrm {d}x\) on \({\mathbb {R}}^{m+1}\) replaced by \(x_{m+1}^{\beta }\mathrm {d}x\), yields the following.
Lemma 3.3
If \(u \in {\mathcal {D}}_{\beta }\), then
for all \(\phi \in C^{\infty }_0({\mathcal {O}};{\mathbb {R}}^n)\). If \(u \not \in {\mathcal {D}}_{\beta }\), then there exists \(\phi \in C^{\infty }_0({\mathcal {O}};{\mathbb {R}}^n)\) such that the function \(t \mapsto I^{\beta }(\pi _N(u + t\phi ))\) is not differentiable at 0.
Consequently, we may define intrinsic fractional harmonic maps as follows.
Definition 3.4
Let \(\beta \in (-1,1)\) and \(u \in T({\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N))\). If \({\varLambda }_{\beta }u = 0\) then we say that u is an intrinsic \(\frac{1-\beta }{2}\)-harmonic map.
As discussed in the introduction, intrinsic \(\frac{1-\beta }{2}\)-harmonic maps are the boundary values of free boundary harmonic maps from \({\mathbb {R}}^{m+1}_+\) to N. Such maps may have singularities in \({\mathcal {O}}\) and we may not expect regularity in general. We consider a smaller class of fractional harmonic maps which locally minimise \(I^{\beta }\) in order to obtain partial regularity in \({\mathcal {O}}\).
Definition 3.5
We say that \(u \in T({\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N))\) is an intrinsic locally minimising \(\frac{1-\beta }{2}\)-harmonic map, or a local minimiser of \(I^{\beta }\), if for every compact \(K\subset {\mathcal {O}}\) and every \({\tilde{u}} \in T({\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N))\) with \(u\vert _{{\mathcal {O}}{\backslash } K} = {\tilde{u}}\vert _{{\mathcal {O}}{\backslash } K}\) we have \(I^{\beta }(u) \le I^{\beta }({\tilde{u}})\).
This definition allows us to consider interior regularity for a class of critical points of \(I^{\beta }\) without explicitly specifying boundary conditions. For example, minimisers of \(I^{\beta }\) with respect to Dirichlet or free boundary conditions satisfy the definition. In order to deduce regularity results for u, we analyse their minimal harmonic extensions in more detail. To this end, we make the following definition.
Definition 3.6
Let \(v \in {\dot{W}}_{\beta }^{1,2}({\mathbb {R}}^{m+1}_+;N) \). We say that v is \(E^{\beta }\) minimising, or energy minimising, in \({\mathbb {R}}^{m+1}_+\) relative to \({\mathcal {O}} \subset \partial {\mathbb {R}}^{m+1}_+\), if for every compact \(K\subset {\mathbb {R}}^{m+1}\) with \(K\cap \partial {\mathbb {R}}^{m+1}_+ \subset {\mathcal {O}}\) and for every \(w \in {\dot{W}}_{\beta }^{1,2}({\mathbb {R}}^{m+1}_+;N) \) with \(\left. v\right| _{{\mathbb {R}}^{m+1}_+ {\backslash } K} = \left. w\right| _{{\mathbb {R}}^{m+1}_+ {\backslash } K} \) we have \(E^{\beta }(v) \le E^{\beta }(w)\).
Local minimisers of \(I^{\beta }\) and minimisers of \(E^{\beta }\) relative to \({\mathcal {O}}\) are connected as follows.
Lemma 3.7
Suppose \(u \in T({\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N))\) locally minimises \(I^{\beta }\) in the sense of Definition 4.11 and fix a minimal harmonic map \(v \in {\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N)\) with \(Tv = u\). Then v is a minimiser of \(E^{\beta }\) relative to \({\mathcal {O}}\).
Proof
Let \(K \subset {\mathbb {R}}^{m+1}\) be compact such that the compact set \(K_m:=K \cap \partial {\mathbb {R}}^{m+1}_+ \subset {\mathcal {O}}\) and suppose that \(w \in {\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N)\) satisfies \(v\vert _{{\mathbb {R}}^{m+1}_+{\backslash } K} = w\vert _{{\mathbb {R}}^{m+1}_+{\backslash } K}\). Define \({\tilde{u}}=Tw\) and let \({\tilde{v}}\) be a minimal harmonic map with \(T{\tilde{v}} = {\tilde{u}}\). Since \({\mathcal {O}}\) is open in \(\partial {\mathbb {R}}^{m+1}_+\) and \(K_m \subset {\mathcal {O}}\) is compact we have \(\text {dist}^m(K_m;\partial {\mathcal {O}})>0\), where \(\text {dist}^m\) is the distance in \(\partial {\mathbb {R}}^{m+1}_+\). We can therefore choose an open set \(\tilde{{\mathcal {O}}}\subset {\mathcal {O}}\) with \(K_m\subset \tilde{{\mathcal {O}}}\subset \overline{\tilde{{\mathcal {O}}}}\subset {\mathcal {O}}\). Since \(K_m\) is closed and \(\tilde{{\mathcal {O}}}\) is open we have \(\text {dist}^m(K_m; \partial \tilde{{\mathcal {O}}})>0\) as well. It follows that \(\text {dist}({\mathcal {O}}{\backslash }\overline{\tilde{{\mathcal {O}}}};K) := \kappa >0\), where \(\text {dist}\) is the Euclidean distance in \(\overline{{\mathbb {R}}^{m+1}_+}\). The continuity of the trace operator yields
since \(v = w\) in \({\mathcal {O}}{\backslash }\overline{\tilde{{\mathcal {O}}}}\times (0,\kappa )\), where p is the number from Lemma 2.1 depending on \(\beta \). Since v and \({\tilde{v}}\) are minimal harmonic maps and u is a minimiser of \(I^{\beta }\), we have
as required. \(\square \)
As a consequence of the preceding lemma, we can consider the regularity of minimisers of \(E^{\beta }\) relative to \({\mathcal {O}}\) on relatively open balls (in the Euclidean topology) centred on \({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}\) in order to prove regularity of fractional harmonic maps. Our main results, stated and proved in Sect. 4, constitute an \(\varepsilon \)-regularity and a corresponding partial regularity theorem for minimisers of \(E^{\beta }\) relative to \({\mathcal {O}}\) which translate into the following partial regularity result for fractional harmonic maps.
Theorem 3.8
Fix \({\mathcal {O}}\subset \partial {\mathbb {R}}^{m+1}_+\). When \(m \ge 3\) let \(\beta \in (-1,1)\) and when \(m =2\) let \(\beta \in (-3^{-1},1)\). Suppose \(u \in T({\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N))\) is a local minimiser of \(I^{\beta }\). Then there exists a relatively closed set \({\varSigma }\subset {\mathcal {O}}\) with \({\mathcal {H}}^{m +\beta - 1}({\varSigma }) = 0\) such that \(u \in C^{\infty }( {\mathcal {O}}{\backslash }{\varSigma };N)\).
Proof
Fix a minimal harmonic map \(v \in {\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N)\) with \(Tv = u\). As u is a minimiser of \(I^{\beta }\), Lemma 3.7 implies v is a minimiser of \(E^{\beta }\) relative to \({\mathcal {O}}\). An application of Theorem 4.3 implies the result. \(\square \)
4 Partial regularity of minimisers of \(E^{\beta }\) relative to \({\mathcal {O}}\)
The main result of this section is the following \(\varepsilon \)-regularity theorem for minimisers of \(E^{\beta }\) relative to \({\mathcal {O}}\). To state the theorem and subsequent results we will need the following notation. For a set \({\varOmega }\subset {\mathbb {R}}^{m+1}_+\) we will sometimes split the boundary \(\partial {\varOmega }\) into the (possibly empty) sets \(\partial ^+{\varOmega }= \partial {\varOmega }\cap {\mathbb {R}}^{m+1}_+\) and \(\partial ^0{\varOmega }= \partial {\varOmega }\cap \partial {\mathbb {R}}^{m+1}_+\). Let \(x_0 \in \partial {\mathbb {R}}^{m+1}_+\) and recall the notation \(B^+_R(x_0) = \{x \in {\mathbb {R}}^{m+1}_+{:}\,\vert x - x_0\vert <R\}\).
Theorem 4.1
If \(m \ge 3\), let \(\beta \in (-1,1)\) and if \(m =2\) let \(\beta \in (-3^{-1},1)\). Let \(v \in {\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N)\) be a minimiser of \(E^{\beta }\) relative to \({\mathcal {O}}\), let \(x_0 \in \partial {\mathbb {R}}^{m+1}_+\) and \(B^+_R(x_0)\) be a half-ball with \(R \le 1\) and \(\overline{\partial ^0B^+_R(x_0)}\subset {\mathcal {O}}\). There exists \(\varepsilon = \varepsilon (m,N,\beta )>0\) such that the following holds. If \(R^{1-m-\beta }\int _{B^+_R(x_0)}x_{m+1}^{\beta }\vert \nabla v\vert ^2\mathrm {d}x \le \varepsilon \) then there is a \(\theta = \theta (m,N,\beta ) \in (0,1)\) and a \(\gamma = \gamma (m,N,\beta ) \in (0,1)\) such that \( v \in C^{0,\gamma }(\overline{B^+_{\theta R}(x_0)};N)\). Furthermore, for every \(l \in {\mathbb {N}}\) there is a \(\theta = \theta (m,N,\beta ,l) \in (0,1)\) and a \(\gamma = \gamma (m,N,\beta ,l) \in (0,1)\) such that \(D^{\alpha '}v \in C^{0,\gamma }(\overline{B^+_{\theta R}(x_0)};{\mathbb {R}}^n)\) for every \(\alpha ' \in {\mathbb {N}}_0^{m+1}\) with \(\vert \alpha '\vert \le l\) and \(\alpha '_{m+1} = 0\).
Remark 4.2
Henceforth, we assume the conditions on m and \(\beta \) from Theorem 4.1. We have restricted to considering \(\alpha '\) with \(\alpha '_{m+1} = 0\) as (partial) regularity of these derivatives up to the boundary will yield the desired regularity for fractional harmonic maps stated in Theorem 3.8. The main purpose of the theorem is to provide regularity estimates which are uniform up to \({\mathcal {O}}\); such estimates do not follow from known theory.
Theorem 4.1, combined with the partial regularity theory for harmonic maps yields the following. We use the notation \({\mathcal {H}}^t\) to denote the t-dimensional Hausdorff measure, with respect to the Euclidean metric on \({\mathbb {R}}^{m+1}\), for \(t \ge 0\).
Theorem 4.3
Let \(v \in {\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N)\) be a minimiser of \(E^{\beta }\) relative to \({\mathcal {O}}\). There exists sets \({\varSigma }_{\text {int}}\subset {\mathbb {R}}^{m+1}_+\) and \({\varSigma }_{\text {bdry}} \subset {\mathcal {O}}\) such that the following holds. The set \({\varSigma }_{\text {int}}\) is relatively closed in \({\mathbb {R}}^{m+1}_+\) and has Hausdorff dimension at most \(m-2\). The set \({\varSigma }_{\text {bdry}}\) is relatively closed in \({\mathcal {O}}\) and \({\mathcal {H}}^{m+\beta -1}({\varSigma }_{\text {bdry}}) = 0\). The set \({\varSigma }:= {\varSigma }_{\text {int}} \cup {\varSigma }_{\text {bdry}}\) is relatively closed in \({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}\) and \({\mathcal {H}}^{m +\beta - 1}({\varSigma }) = 0\). Furthermore, we have \(v \in C^{\infty }({\mathbb {R}}^{m+1}_+{\backslash } {\varSigma }_{\text {int}};N)\), \(v \in C^{0,1}_{loc}(({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}){\backslash } {\varSigma };N)\) and for every multi-index \(\alpha ' \in {\mathbb {N}}^{m+1}\) with \(\alpha '_{m+1} = 0\) we have \(D^{\alpha '}v \in C^{0,1}_{loc}(({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}){\backslash } {\varSigma };{\mathbb {R}}^n)\) and \(\nabla D^{\alpha '}v \in L^{\infty }_{loc}(({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}){\backslash } {\varSigma };{\mathbb {R}}^{(m+1)n})\). Finally, for every \(\alpha '\in {\mathbb {N}}^{m+1}_0\) with \(\alpha '_{m+1} = 0\), we have \(x_{m+1}^{\beta }\partial _{m+1}D^{\alpha '}v \in C^{0, {\gamma }}_{loc}(({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}){\backslash } {\varSigma };{\mathbb {R}}^{(m+1)n})\) for some \( {\gamma } = {\gamma }(m,N,\beta , \alpha ')\in (0,1)\).
Remark 4.4
The existence and properties of \({\varSigma }_{\text {int}}\) follows from the theory of Schoen and Uhlenbeck [37].
4.1 Euler–Lagrange and stationary equations for minimisers
Minimisers of \(E^{\beta }\) relative to \({\mathcal {O}}\) are critical points of \(E^{\beta }\) with respect to outer and inner variations, including those which vary their boundary data in \({\mathcal {O}}\). As a consequence, they satisfy two systems of partial differential equations which we describe presently.
Let \(\psi \in {\mathcal {D}}_+({\mathbb {R}}^{m+1}_+;{\mathbb {R}}^n)\), as defined in Sect. 2.1, with \(\psi (\cdot ,0) \in C^{\infty }_0({\mathcal {O}};{\mathbb {R}}^n)\). For sufficiently small t we define an outer variation of v by \(v_t = \pi _N(v + t\psi ) \in N\), where \(\pi _N\) is the nearest point projection onto N. Critical points of \(E^{\beta }\) with respect to variations of the form \(v_t\) satisfy
Note that there is a Neumann-type boundary condition implicit in (4.1). In particular, if v is sufficiently smooth in \( {\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}\) we have
A (weakly) harmonic map satisfying (4.1) is said to be (weakly) harmonic with respect to the Neumann type boundary condition (4.2).
Define \({\varPhi }_t(x) = x + t\phi (x)\) for \(x \in \overline{{\mathbb {R}}^{m+1}_+}\), where \(\phi \in {\mathcal {D}}_+({\mathbb {R}}^{m+1}_+;{\mathbb {R}}^{m+1})\) is such that \(\phi (\cdot ,0) \in C^{\infty }_0({\mathcal {O}};\partial {\mathbb {R}}^{m+1}_+)\) and |t| is small enough to make \({\varPhi }_t\) into a diffeomorphism of \(\overline{{\mathbb {R}}^{m+1}_+}\) with \({\varPhi }_t({\mathcal {O}})\subset {\mathcal {O}}\). We say \(v \in {\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N)\) is a critical point of the Dirichlet energy corresponding to inner variations \(v_t := v\circ {\varPhi }_t\), or variations of the independent variable, if v satisfies
for every \(\phi \) as above. A weakly harmonic map with respect to the Neumann type boundary condition (4.2) which satisfies (4.3) for every \(\phi \in {\mathcal {D}}_+({\mathbb {R}}^{m+1}_+;{\mathbb {R}}^{m+1})\) with \(\phi (\cdot ,0) \in C^{\infty }_0({\mathcal {O}};\partial {\mathbb {R}}^{m+1}_+)\) is called weakly stationary harmonic, or stationary harmonic, with respect to the Neumann type boundary condition (4.2).
4.2 Energy monotonicity
Stationary harmonic maps satisfy a monotonicity formula for an appropriately scaled version of the energy over balls with closure in \({\mathbb {R}}^{m+1}_+\). This property was proved by Schoen and Uhlenbeck for energy minimisers, see [37, Proposition 2.4], and Price, see the remark after Theorem 1 in [32], for stationary harmonic maps.
As a consequence of (4.3), we show that stationary harmonic maps with respect to the Neumann-type boundary condition (4.2) satisfy a similar monotonicity formula on half-balls \(B^+_{\rho }(y)\) with centre y in \({\mathcal {O}}\) and which satisfy \(\overline{\partial ^0 B^+_{\rho }(y)} = \overline{B^m(y)} \subset {\mathcal {O}}\). Moreover, we state a version of the formula for balls with closure contained in \({\mathbb {R}}^{m+1}_+\), giving an explicit expression for the factors that the constants involved depend upon.
Lemma 4.5
Suppose \(v \in {\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N)\) is a weakly stationary harmonic map with respect to the Neumann-type boundary condition (4.2). Suppose y in \({\mathcal {O}}\) and consider \(B^+_R(y)\) with \(\overline{\partial ^0 B^+_{R}(y)}\subset {\mathcal {O}}\). Then
whenever \(0 \le s \le r\le R\) and therefore \(\rho \mapsto \rho ^{1-m - \beta }\int _{B^+_{\rho }(y)}x_{m+1}^{\beta }\left| \nabla v\right| ^2\mathrm {d}x\) is a non-decreasing function of \(\rho \) for \(0<\rho \le R\).
Proof
The proof is analogous to that of the monotonicity formula for stationary harmonic maps. We follow [39, Section 2.4] and [29, Lemma 3.3]; we test (4.3) with \(\phi (x) = (x-y) \eta (x) \), where \(\eta \in C^{\infty }_0(B_{\rho }(y))\), which yields
Let \(\chi \in C_0^{\infty }({\mathbb {R}};[0,1])\) with \(\chi (s) \equiv 1\) for \(s \ge 1\) and \(\chi (s) \equiv 0\) for \(s\le \frac{1}{2}\). The smooth functions defined by \(\eta _j(x) =\chi (j(\rho - \vert x - y \vert ))\) are admissible choices for \(\eta \) in (4.4) and \(\{\eta _j\}_{j \in {\mathbb {N}}}\) converges pointwise to the indicator function of \(B^+_{\rho }(y)\). We substitute \(\eta _j\) for \(\eta \) in (4.4) and take the limit as \(j \rightarrow \infty \), using Lebesgue’s Dominated Convergence and Differentiation Theorems, to see that
for almost every \(\rho >0\), where \(\mathrm {d}S\) is the Lebesgue measure on \(\partial B_{\rho }(y)\). Multiplying the above by the factor \(-\rho ^{-(\beta +m)}\) and bearing in mind that \(\frac{\mathrm {d}}{\mathrm {d}\rho }\int _{B^+_{\rho }(y)}x_{m+1}^{\beta }\left| \nabla v\right| ^2\mathrm {d}x = \int _{\partial ^+ B^+_{\rho }(y)}x_{m+1}^{\beta }\left| \nabla v\right| ^2\mathrm {d}S(x)\) for almost all \(\rho > 0\), we find
for almost every \(\rho >0\). Integrating between \(0<s<r\) concludes the proof. \(\square \)
Remark 4.6
A consequence of Lemma 4.5 is that we can define the density function
for every \(y \in {\mathcal {O}}\), analogously to Definition 1 in Section 2.5 of [39]. Using Lemma 4.5 we deduce \({\varTheta }^{\beta }_v\) is upper semi-continuous in \({\mathcal {O}}\) for any map v which is weakly stationary harmonic with respect to the Neumann-type boundary condition (4.2).
The following version of the energy monotonicity formula is due to Grosse-Brauckmann [16, Theorem 1]. We do not give a proof, but remark that the explicit form of the constant in the forthcoming formula can be determined using the method of proof of Lemma 4.5.
Lemma 4.7
Suppose \(v \in {\dot{W}}_{\beta }^{1,2}({\mathbb {R}}^{m+1}_+;N)\) is a minimiser of \(E^{\beta }\) relative to \({\mathcal {O}}\). Fix a ball \(B_{\rho _0}(y)\) with \(\overline{B_{\rho _0}(y)}\subset {\mathbb {R}}^{m+1}_+\) for some \(\rho _0>0\) and suppose r and s satisfy \(0<s<r<\rho _0\). Then
and therefore, for \(0<\rho <\rho _0\), \(\rho \mapsto e^{ \rho C\vert \beta \vert }\rho ^{1-m}\int _{B_{\rho }(y)}x_{m+1}^{\beta }\left| \nabla v\right| ^2\mathrm {d}x\) is a non-decreasing function of \(\rho \) where \(C = (y_{m+1} - \rho _0)^{-1} = \left( \text {dist}(B_{\rho _0}(y),\partial {\mathbb {R}}^{m+1}_+)\right) ^{-1}\).
4.3 A modified lemma of Morrey
In order to prove Theorem 4.21, in analogy with the regularity theory of harmonic maps, we will show that the re-scaled, scale-invariant energies in the monotonicity formulas in Sect. 4.2 decay slightly faster than implied by the Lemmata as the radius decreases. This will permit the application of a well-known lemma of Morrey, see [29, Lemma 2.1] for example, which is used to derive Hölder continuity from sufficiently fast energy decay. We will reduce the hypothesis of this lemma to similar hypothesis for the re-scaled energies from the monotonicity formula. To this end, we introduce a class of ball with closure in \({\mathbb {R}}^{m+1}_+\) on which the metrics \(x_{m+1}^{\alpha }\delta _{ij}\), discussed in Sect. 3 and corresponding weights \(x_{m+1}^{\beta }\) are uniformly equivalent to the Euclidean metric and 1 respectively. We also introduce classes of balls and half-balls contained in the interior of a given larger half-ball \(B_R^+(x_0)\) for \(R>0\) and \(x_0 \in \partial {\mathbb {R}}^{m+1}_+\).
Define
for \(\theta \ge 2\). Then \({\mathcal {B}}_{\theta } \subset {\mathcal {B}}\) and \({\mathcal {B}}_{2} = {\mathcal {B}}\). We further define
omitting the subscript \(\theta \) in the case \(\theta = 2\), and let
Observe that, on any \(B_{\rho }(y) \in {\mathcal {B}}\), we can choose constants \(c,C,c_0\) and \(C_0\) independently of \(\beta \) such that for every \(x \in B_{\rho }(y)\) and \(\beta \in (-1,1)\) we have
Lemma 4.8
Let \(\gamma >0\), \(x_0 \in \partial {\mathbb {R}}^{m+1}_+\), \(a > 0\), \(\theta _1 \ge 2\) and \(\theta _2 \le \frac{1}{2}\). Define \(\theta = \frac{\theta _2}{2\theta _1}\). Then there exists a constant \(C_0 = C_0(m,\gamma ,\theta _1,\beta )\) such that if \(v \in {W}^{1,2}_{\beta }(B^+_{R}(x_0);{\mathbb {R}}^n)\) with
for every \(B = B^+_{r}(y) \in {\mathcal {B}}^+(x_0,R,\theta _2R)\) and every \( B = B_{r}(y) \in {\mathcal {B}}_{\theta _1}(x_0,R,\theta _2R)\), then for almost every \(x_1,x_2 \in B^+_{\theta R}(x_0)\),
Proof
Let \(B_{r}(y)\subset B_{2\theta R}(x_0)\) with \(y_{m+1} \ge 0\), \(y \in B_{\theta R}(x_0)\) and \(r \le \theta R\), for a \( \gamma \in (0,1)\), an \(a>0\) and \(\theta \) as specified. Such a \(B_{r}(y)\) must satisfy either \(B_{r}(y) \in {\mathcal {B}}_{\theta _1}(x_0,2\theta R,\theta R)\) or \(B_{r}(y) \not \in {\mathcal {B}}_{\theta _1}(x_0,2\theta R,\theta R)\). We consider these cases in turn and we work with the even reflection of v with respect to \(\partial {\mathbb {R}}^{m+1}_+\), which we do not relabel and which is in \(W^{1,2}_{\beta }(B_{R}(x_0);{\mathbb {R}}^n)\).
Suppose \(B_{r}(y) \in {\mathcal {B}}_{\theta _1}(x_0,2\theta R,\theta R)\) with \(r \le \theta R\). Then \(B_{r}(y) \in {\mathcal {B}}_{\theta _1}(x_0,R,\theta _2R)\) and an application of Hölder’s inequality and the assumptions of the lemma yields
Now suppose \(B_{r}(y) \not \in {\mathcal {B}}_{\theta _1}(x_0,2\theta R,\theta R)\) and \(r \le \theta R\). In this case, since \(B_{r}(y)\subset B_{2\theta R}(x_0)\) and \(y \in B_{\theta R}(x_0)\) by assumption, we must have \(y_{m+1} < \theta _1r\). Hence \(y_{m+1} - r<\zeta r\), where \(\zeta \ge 1\) is such that \(\theta _1 = \zeta +1\), and thus \(B_{r}(y)\subset B_{ (2+ \zeta ) r }(y_0)\) where \(y_0 = y - (0,y_{m+1})\). We observe that \( B^+_{ (2+ \zeta ) r }(y_0) \in {\mathcal {B}}^+(x_0,R,\theta _2R)\). Therefore, defining \(s = (2+ \zeta ) r\) and using the assumptions of the lemma, the symmetry of v and applying Hölder’s inequality, we find
Since v is even with respect to \(\partial {\mathbb {R}}^{m+1}_+\), we deduce that either (4.8) or (4.9) holds on any \(B_{r}(y)\subset B_{2\theta R}(x_0)\) with \(y \in B_{\theta R}(x_0)\) and \(r \le \theta R \). Hence we have established that the hypothesis of the decay lemma of Morrey hold on \( B_{2\theta R}(x_0)\), see [29, Lemma 2.1]. An application of this lemma concludes the proof. \(\square \)
4.4 Interior estimates for Hölder continuity
Using the regularity theory of Schoen and Uhlenbeck [37] and Schoen [38], we show that minimisers \(v \in {\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N)\) of \(E^{\beta }\) relative to \({\mathcal {O}}\) essentially satisfy (4.7) in Lemma 4.8, provided the scale-invariant energy \(R^{1-m-\beta }\int _{B^+_{R}(x_0)}x_{m+1}^{\beta }\vert \nabla v\vert ^2\mathrm {d}x\) is sufficiently small. To this end, we show that the preceding scale-invariant energy \(B^+_R(x_0)\) controls the scale-invariant Euclidean energy on a class of ball with closure in \(B^+_R(x_0)\). We also recall the relevant theory from [37, 38, Sections 1, 2 and 3], stating the results in our context with slightly different notation. Our goal is to prove the following.
Lemma 4.9
Suppose \(v \in {\dot{W}}_{\beta }^{1,2}({\mathbb {R}}^{m+1}_+;N)\) is a minimiser of \(E^{\beta }\) relative to \({\mathcal {O}}\). Let \(B^+_R(x_0)\) be a half-ball with \(R \le 1\) and \(\overline{\partial ^0B^+_{R}(x_0)}\subset {\mathcal {O}}\). There exists an \(\varepsilon _0=\varepsilon _0(m,N)>0\), a \(\theta =\theta (m,N)\ge 2\) and a positive \(C =C(m,N)\) such that if
then
on every \(B_{r}(y) \in {\mathcal {B}}_{\theta }(x_0,R,\frac{R}{3})\) for \(0< \rho \le r\) and a \(\gamma = \gamma (m,N)\in (0,1)\).
To establish the preceding lemma, we observe the following relationship between the scale-invariant energy on \(B^+_R(x_0)\) and the scale-invariant Euclidean energy on a class of ball with closure in \(B^+_R(x_0)\).
Lemma 4.10
Suppose \(v \in {\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N)\) is a minimiser of \(E^{\beta }\) relative to \({\mathcal {O}}\). Let \(B^+_{R}(x_0)\) be a half-ball with \(\overline{\partial ^0B^+_{R}(x_0)}\subset {\mathcal {O}}\) and suppose \(B_{\rho }(y) \in {\mathcal {B}}(x_0,R,\frac{R}{3})\). Then there is a constant \(C = C(m)\) such that
Proof
Notice that any ball \(B_{\rho }(y) \in {\mathcal {B}}\) satisfies \(B_{\rho }(y) \subset B_{\frac{y_{m+1}}{2}} (y)\) so we can choose the scaling factor \(e^{\frac{2\vert \beta \vert \rho }{y_{m+1}}}\) in Lemma 4.7. Furthermore, \(e^{\frac{2\vert \beta \vert \rho }{y_{m+1}}}\le e\) since \(y_{m+1} \ge 2\rho \) and \(\beta \in (-1,1)\). Hence, using (4.6) and applying Lemma 4.7, we find
Let \(y = (y_1,\ldots ,y_{m+1})\) and \(y^+ = (y_1,\ldots ,y_m,0)\). Note that \(B_{\frac{y_{m+1}}{2}}(y) \subset B^+_{\frac{3y_{m+1}}{2}}(y^+)\) and, since \(B_{\rho }(y) \in {\mathcal {B}}\left( x_0,R,\frac{R}{3}\right) \), we have \(y \in B^+_{\frac{R}{3}}(x_0)\) and \( B^+_{\frac{3y_{m+1}}{2}}(y^+) \subset B^+_{\frac{R}{2}}(y^+) \subset B^+_{R}(x_0)\). Using these facts we apply Lemma 4.5 to see that
where \(C = C(m)\). The combination of (4.12) and (4.13) yields (4.11). \(\square \)
The maps considered in [37, 38] belong to
for open \({\varOmega }\subset {\mathbb {R}}^{m+1}_+\). Consider the compact Riemannian manifold \(\overline{B_1(0)}\) with metric \({\tilde{g}}\). Recall from the introduction that the Dirichlet energy functional on \(\overline{B_1(0)}\) is given by
A minimiser of \({E}_{{\tilde{g}}}\) with fixed boundary data is defined in Section 1 of [37] as follows.
Definition 4.11
Any \(v \in W^{1,2}(B_1(0);N)\) is an \({E}_{{\tilde{g}}}\) minimising map if it satisfies \({E}_{{\tilde{g}}}(v) \le {E}_{{\tilde{g}}}(w)\) for any \(w \in W^{1,2}( {B_1(0)};N)\) with \(v - w \in W^{1,2}_0(\overline{B_1(0)};{\mathbb {R}}^n)\).
The metric \({\tilde{g}}\) is assumed to be of class \(C^2\) on \(\overline{ B_1(0)}\). For \({\varLambda }>0\) denote by \({\mathscr {E}}_{{\varLambda }}\) the class of functionals \({E}_{{\tilde{g}}}\) on \(\overline{ B_1(0)}\) with metric \({\tilde{g}}\) such that \({\tilde{g}}_{ij}(0) = \delta _{ij}\) and
If v is \({E}_{{\tilde{g}}}\)-minimising with \({E}_{{\tilde{g}}} \in {\mathscr {E}}_{{\varLambda }}\) then we say \(v \in {\mathscr {H}}_{{\varLambda }}\).
Schoen and Uhlenbeck [37] proved their \(\varepsilon \)-regularity theorem for minimisers of functionals of the form \({\tilde{E}}_{{\tilde{g}}} + F\), where F gives rise to terms in the Euler–Lagrange equations which are lower order than those coming from the energy. We state the result of their theorem with \(F = 0\). The following Lemma is Theorem 3.1 in [37].
Lemma 4.12
There exists \(\varepsilon =\varepsilon (m,N) > 0\) such that if \(v \in {\mathscr {H}}_{{\varLambda }}\), \({\varLambda }\le \varepsilon \) and \(\int _{B_{1}(0)}\left| \nabla v\right| ^2\mathrm {d}x \le \varepsilon \), then v is Hölder continuous in \(B_{\frac{1}{2}}(0)\) and
for constants \(C = C(m,N)\) and \(\gamma = \gamma (m,N)\in (0,1)\) and every \(x_1,x_2 \in B_{\frac{1}{2}}(0)\).
It is well known that continuous weakly harmonic maps are smooth, see [22] for example. It is more readily shown that Hölder continuous harmonic maps are smooth; this is the content of the following lemma, which is Lemma 3.1 of [38].
Lemma 4.13
Consider a ball \(B_r(y) \subset {\mathbb {R}}^{m+1}_+\) and suppose \(v \in W^{1,2}(B_r(y);N)\) is a weakly harmonic map which is Hölder continuous on \(B_r(y)\). Then v is smooth on \(B_r(y)\).
The final lemma we will need is Theorem 2.2 from [38].
Lemma 4.14
Let \(v \in C^2(B_{r}(0);N)\) and \({\tilde{g}}\) be a Riemannian metric on \(B_{r}(0)\). Suppose v is harmonic with respect to \({\tilde{g}}\) in \(B_r(0)\) and \({\tilde{g}}\) satisfies \(\left| \partial _k {\tilde{g}}_{ij} \right| \le {\varLambda }r^{-1}\) for \(i,j,k = 1,\ldots , m+1\) and \({\varLambda }^{-1}(\delta _{ij}) \le ({\tilde{g}}_{ij}) \le {\varLambda }(\delta _{ij})\) in the sense of tensors, where \(\delta _{ij} = 1\) when \(i = j\) and \(\delta _{ij} = 0\) otherwise. Then there exists an \(\varepsilon = \varepsilon ({\varLambda },m,N)>0\) such that if
then
for a constant \(C = C({\varLambda },m,N)\).
Proof of Lemma 4.9
Suppose \(R^{1-m-\beta }\int _{B^+_R(x_0)}x_{m+1}^{\beta }\vert \nabla v\vert ^2\mathrm {d}x \le \varepsilon _0\) for an \(\varepsilon _0>0\) to be chosen small and let \(\varepsilon \) be the number from Lemma 4.12.
Recall the metric g given in Euclidean coordinates by \(g:= x_{m+1}^{\alpha }\delta _{ij}\) and define \({\hat{g}}\) on \(\overline{B_1(0)}\) by
The energy corresponding to \({\hat{g}}\) is
for maps \({\hat{v}} \in W^{1,2}({B_1(0)};{\mathbb {R}}^n)\). Since \(B_r(y) \in B_{\theta }(x_0,R,\frac{R}{3}) \) for a \(\theta \ge 2\) to be chosen, we have \(B_r(y) \in {\mathcal {B}}\). Hence, using (4.6) and noting that \(\beta = \alpha \left( \frac{m-1}{2}\right) \in (-1,1)\), we find constants c, C depending only on m such that
We note that \(\partial _k{\hat{g}} = 0\) for \(k \ne m+1\) and, again using (4.6), we calculate
where C is chosen independently of \(\alpha \). Hence, if we set \(\theta = \theta (m,N) \ge \max \left\{ 2, (m+1)C \varepsilon ^{-1}\right\} \) then we conclude that
We assume the preceding choice of \(\theta \) henceforth so that (4.17) holds on any \(B_{r}(y) \in {\mathcal {B}}_{\theta }(x_0,R,\frac{R}{3})\).
Define \(v_{r,y}(x) = v(rx+y)\) for \(x \in B_1(0)\). Lemmata 2.1 and 2.3 imply \({\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N)\hookrightarrow {W}^{1,2}(B_{r}(y);N)\) for every \(B_{r}(y)\) with \(\overline{B_{r}(y)}\subset {\mathbb {R}}^{m+1}_+\), regardless of \(\beta \in (-1,1)\). A change of variables then yields \(v_{r,y} \in {W}^{1,2}(B_{1}(0);N)\). Furthermore, since v is a minimiser of \(E^{\beta }\) relative to \({\mathcal {O}}\) we readily calculate that \(v_{r,y}\) is a minimiser of \(E_{{\hat{g}}}\) in the sense of Definition 4.11 on \(B_1(0)\), that is, \(v_{r,y}\) minimises \(E_{{\hat{g}}}\) among all maps in \({W}^{1,2}(B_{1}(0);N)\) with the same boundary values as \(v_{r,y}\).
Our considerations so far imply that if \(B_{r}(y) \in {\mathcal {B}}_{\theta }(x_0,R,\frac{R}{3})\), for our preceding choice of \(\theta \), then \(v_{r,y} \in {\mathscr {H}}_{\varepsilon }\). Let C be the constant from Lemma 4.10 and suppose that \(\varepsilon _0 \le C^{-1}\varepsilon \). An application of this lemma, combined with a change of variables, yields
This holds for every \(B_{r}(y) \in {\mathcal {B}}_{\theta }\left( x_0,R,\frac{R}{3}\right) \). We may therefore apply Lemma 4.12 to \(v_{r,y}\) to deduce that it is Hölder continuous in \( B_{\frac{1}{2}}(0)\). Re-scaling implies v is Hölder continuous in every \(B_{\frac{r}{2}}(y) \in {\mathcal {B}}_{\theta }(x_0,R,\frac{R}{3})\), or equivalently v is Hölder continuous in every \(B_{r}(y) \in {\mathcal {B}}_{2\theta }(x_0,R,\frac{R}{3})\).
Since v is weakly harmonic in \({\mathbb {R}}^{m+1}_+\) with respect to the metric represented by \(x_{m+1}^{\alpha }\delta _{ij}\) (where \(\beta = \alpha \frac{m-1}{2}\)), it is weakly harmonic with respect to \(x_{m+1}^{\alpha }\delta _{ij}\) on every \(B_{r}(y)\) with \(\overline{B_{r}(y)}\subset {\mathbb {R}}^{m+1}_+\). Thus it follows from Lemma 4.13 that v is smooth in each \(B_{r}(y) \in {\mathcal {B}}_{2\theta }(x_0,R,\frac{R}{3})\), which holds if and only if each \(v_{r,y}\) corresponding to such a \(B_r(y)\) is smooth in \(B_{1}(0)\). We further deduce that \(v_{r,y}\) is harmonic in \(B_1(0)\) with respect to \({\hat{g}}\) using the chain rule. Moreover, it follows from (4.16) and (4.17) that \({\hat{g}}\) satisfies the assumptions required of the metric in Lemma 4.14. We combine (4.16) with (4.18) to see that
In addition to our previous stipulation for \(\varepsilon _0\), we further require that \(\varepsilon _0 \le \frac{\varepsilon _1}{C}\) where \(\varepsilon _1\) is the number from Lemma 4.14. We apply this lemma, recalling \(\beta = \alpha (\frac{m-1}{2})\) and \(B_r(y) \in {\mathcal {B}}\), to see that
As a result, for any \(\sigma \in (0,\frac{1}{2}]\) we have
Choose \(\sigma \) such that \(\sigma ^2 \le \frac{1}{2C}\) and let \(\rho \le r\). Then
on every \(B_{r}(y) \in {\mathcal {B}}_{2\theta }(x_0,R,\frac{R}{3})\). The map \(\rho \mapsto e^{\frac{\vert \beta \vert \rho }{y_{m+1}-r}} \rho ^{1-m} \int _{B_{ \rho }(y)} \left( x_{m+1}y_{m+1}^{-1}\right) ^{\beta }\vert \nabla v \vert ^2 \mathrm {d}x\) is non-decreasing in \(\rho \) for \(\rho \le r\) as a result of Lemma 4.7. Lemma 8.23 of [15] hence implies
for every \(\rho \le r\) on every \(B_{r}(y) \in {\mathcal {B}}_{2\theta }(x_0,R,\frac{R}{3})\). It follows that
on every \(B_{r}(y) \in {\mathcal {B}}_{2\theta }(x_0,R,\frac{R}{3})\) for \(0< \rho \le r\). This concludes the proof. \(\square \)
4.5 A modified lemma of Luckhaus
Here we begin our construction of comparison maps. We prove a partial extension (to the particular case of our degenerate/singular metrics \(x_{m+1}^{\alpha }\delta _{ij}\)) of a lemma of Luckhaus, Lemma 3 in [23], as presented in Lemma 1 Section 2.6 of [39].
Let \({\mathbb {S}}^{m}\subset {\mathbb {R}}^{m+1}\) denote the m dimensional unit sphere, centred at the origin and equipped with the metric induced by the Euclidean metric on \({\mathbb {R}}^{m+1}\). Define \({\mathbb {S}}^{m}_+ = {\mathbb {S}}^{m}\cap {\mathbb {R}}^{m+1}_+\) with the metric induced from \({\mathbb {S}}^{m}\). We let \(\omega \) denote a point in \({\mathbb {S}}^{m}\subset {\mathbb {R}}^{m+1}\) or \({\mathbb {S}}^{m}_+ \subset {\mathbb {R}}^{m+1}_+\)and write \(\mathrm {d}\omega \) for the volume element corresponding to the induced metric. Recall the notation \(\partial ^+ {\varOmega }= \partial {\varOmega }\cap {\mathbb {R}}^{m+1}_+\) for \({\varOmega }\subset {\mathbb {R}}^{m+1} \) and \(Q_r(y) = \{ x \in {\mathbb {R}}^{m+1}{:}\,\vert x_i - y_i \vert < r, i = 1,\ldots ,m +1 \}\) for \(y \in {\mathbb {R}}^{m+1}\). We also write \(Q_r^+(y) = Q_r(y)\cap {\mathbb {R}}^{m+1}_+\) for \(y \in \partial {\mathbb {R}}^{m+1}_+\).
In order to state the modified Luckhaus lemma precisely we introduce the notion of a Sobolev space for functions whose domain is either \({\mathbb {S}}^{m}\) or \({\mathbb {S}}^m_+\).
Definition 4.15
Let \(\varepsilon >0\) and \(\rho >0\). Suppose \(S=\rho {\mathbb {S}}^{m}\) and \(V_{\varepsilon } = B_{\rho +\varepsilon }(0){\backslash } B_{\rho -\varepsilon }(0)\) or \(S=\rho {\mathbb {S}}^{m}_+\) and \(V_{\varepsilon } = B^+_{\rho +\varepsilon }(0){\backslash } B^+_{\rho -\varepsilon }(0)\). An element \(v \in L^2_{\beta }(S;{\mathbb {R}}^n)\) is said to be in \(W^{1,2}_{\beta }(S;{\mathbb {R}}^n)\) if the map \(v(\rho \frac{x}{\vert x\vert }) \in W^{1,2}_{\beta }(V_{\varepsilon };{\mathbb {R}}^n)\) for some \(\varepsilon >0\). An element \(v \in L^2_{\beta }(S\times [a,b];{\mathbb {R}}^n)\), with \(a<b\) real numbers, is said to be in \(W^{1,2}_{\beta }(S\times [a,b];{\mathbb {R}}^n)\) if the map \(v(\rho \frac{x}{\vert x\vert },s) \in W^{1,2}_{\beta }(V_{\varepsilon }\times [a,b];{\mathbb {R}}^n)\) for some \(\varepsilon >0\). If \(N\subset {\mathbb {R}}^n\) is compact, we say v is in \(W^{1,2}_{\beta }(S;N)\) or \( W^{1,2}_{\beta }(S\times [a,b];N)\) if v is in \(W^{1,2}_{\beta }(S;{\mathbb {R}}^n)\) or \(W^{1,2}_{\beta }(S\times [a,b];{\mathbb {R}}^n)\) respectively and \(v(x) \in N\) for almost every \(x \in S\).
We now state our version of the Luckhaus lemma.
Lemma 4.16
Let \(m \ge 2\) and \(\beta \in (-1,1)\). Let N be a compact subset of \({\mathbb {R}}^n\) and suppose \(u,v \in {W}^{1,2}_{\beta }({\mathbb {S}}^{m}_+ ;N)\). Then for all \(\varepsilon \in (0,1)\) there is a \(w \in {W}^{1,2}_{\beta }({\mathbb {S}}^{m}_+\times [0,\varepsilon ] ;{\mathbb {R}}^n)\) such that w agrees with u on \({\mathbb {S}}^{m}_+ \times \{0\}\) and v on \({\mathbb {S}}^{m}_+ \times \{\varepsilon \}\) in the sense of traces and which satisfies the following. Let \({\overline{D}}\) be the gradient on \({\mathbb {S}}^{m}_+\times [0,\varepsilon ]\) and D the gradient on \({\mathbb {S}}^{m}_+\). Then \(w = w(\omega ,s)\) satisfies
where \(C_1 = C_1(m,\beta )\). Furthermore, w satisfies
for almost every \((\omega ,s) \in {\mathbb {S}}^{m}_+\times [0,\varepsilon ]\) where \(C_2 = C_2(m,\beta )\) and q satisfies the following. If \(\beta \in (-1,0]\) then (4.21) holds for \(q = 2\). If \(\beta \in (0,1)\), for any \(p \in (1,\frac{2}{1+\beta })\), there exists \(q \in \{2,p\}\) such that (4.21) holds.
Our proof of Lemma 4.16 follows the proof, given in Section 2.12.2 of [39], of Lemma 1 in Section 2.6 of [39].
4.6 Absolute continuity properties of functions in \({W}^{1,2}_{\beta }\)
We recall the discussion in [39, Section 2.12.1] regarding the absolute continuity properties of \(W^{1,p}\) functions, which are inherited by \({W}^{1,2}_{\beta }\) functions in view of Lemma 2.1. Let \({\mathcal {H}}^t\) denote the t-dimensional Hausdorff measure with respect to the Euclidean metric. Consider a rectangle \(Q \subset \overline{{\mathbb {R}}^{m+1}_+}\) of the form \(Q = [a_1,b_1] \times \cdots \times [a_{m+1},b_{m+1}]\) where \(a_i < b_i\). Suppose \(v \in {W}^{1,2}_{\beta }(Q;{\mathbb {R}}^n)\) with \(\beta \in (-1,1)\). It follows from Lemma 2.1 that if \(a_{m+1} > 0\) then \(Q \subset {\mathbb {R}}^{m+1}_+\) and \(v\vert _{Q} \in {W}^{1,2}(Q;{\mathbb {R}}^n)\). Lemma 2.1 also implies that if \(a_{m+1} = 0\) then \(v\vert _Q \in {W}^{1,p}(Q;{\mathbb {R}}^n)\) for \(p = p(\beta )\). Hence, by Lemma 3.1.1 and Theorem 3.1.8 in [28], if \(a_{m+1} \ge 0\), we may infer the existence of a representative \({\hat{v}}\) of v such that, for each \(i = 1,\ldots ,m+1\), \({\hat{v}}(x_1,\ldots ,x_{i-1},x_i,x_{i+1},\ldots ,x_{m+1})\) is an absolutely continuous function of \(x_i\) for almost all fixed values of \(x_1,\ldots ,x_{i-1},x_{i+1},\ldots ,x_{m+1}\) with respect to the m dimensional Hausdorff measure \({\mathcal {H}}^m\) on \([a_1,b_1] \times \cdots \times [a_{i-1},b_{i-1}]\times [a_{i+1},b_{i+1}] \times \cdots \times [a_{m+1},b_{m+1}]\). The classical partial derivatives \(\frac{\partial {\hat{v}}}{\partial x_i}\) agree almost everywhere with the weak derivatives \(\frac{\partial v}{\partial x_i}\). Furthermore, for any closed subset N of \({\mathbb {R}}^n\), if \(v(x) \in N\) for almost every x then it is possible to choose \({\hat{v}}(x) \in N\) for every \(x \in {\mathbb {R}}^{m+1}_+\).
4.7 Proof of Lemma 4.16
Proof of Lemma 4.16
We follow the proof, given in Section 2.12.2 of [39], of Lemma 1 in Section 2.6 of [39]. Throughout, C denotes a constant only depending on m and \(\beta \).
Suppose \(u,v \in {W}^{1,2}_{\beta }({\mathbb {S}}^{m}_+ ;N)\). We reflect u and v evenly in \(\partial {\mathbb {R}}^{m+1}_+\), without relabelling, to get \(u,v \in {W}^{1,2}_{\beta }({\mathbb {S}}^{m} ;N)\) and choose extensions of u and v to \({\mathbb {R}}^{m+1}{\backslash }\{0\}\) which are homogeneous of degree zero with respect to the origin. Then we choose representatives of these extensions which satisfy the absolute continuity properties described in Sect. 4.6 on \(\overline{Q_1(0)}\). We will denote the representatives of the extensions of u and v by \({\hat{u}}\) and \({\hat{v}}\) respectively. Then \({\hat{u}}(\rho \omega ) = {\hat{u}}(\omega )\), \({\hat{v}}(\rho \omega ) = {\hat{v}}(\omega )\) for almost every \(\rho > 0\) and \(\omega \in {\mathbb {S}}^{m}\). Moreover, we have the identity \(\nabla {\hat{u}} =\vert x\vert ^{-1}\nabla {\hat{u}}(\omega (x)) = Du(\omega (x))\) for \(\omega (x) =\vert x\vert ^{-1}x\), where D is the gradient on \({\mathbb {S}}^m_+\) and \(\nabla \) is the gradient on \({\mathbb {R}}^{m+1}_+\). We therefore calculate
and
Let \(\varepsilon \in (0,\frac{1}{8})\) and define the closed rectangles \(Q_{i,\varepsilon } = [i_1\varepsilon , (i_1+ 1)\varepsilon ] \times \cdots \times [i_{m+1}\varepsilon , (i_{m+1}+ 1)\varepsilon ]\) for \(i = (i_1, \ldots , i_{m+1}) \in {\mathbb {Z}}^{m+1}\). Fix \(\varepsilon \in (0,\frac{1}{8})\) arbitrarily henceforth. Let \(F^l\) denote any l-dimensional face of a \(Q_{i,\varepsilon } \). We define
In addition, we write \(x + {\mathcal {F}}_i^l \) to denote the collection of the translations of all faces in \({\mathcal {F}}_i^l \) by \(x\in {\mathbb {R}}^{m+1}\).
Consider a non-negative, measurable function \(f{:}\,\overline{Q_1(0)} \rightarrow {\mathbb {R}}\) which is even with respect to the hyperplane \(\partial {\mathbb {R}}^{m+1}_+\). Invoking [39, Section 2.12.2] estimate (3), which is a consequence of Chebychev’s inequality and Fubini’s theorem, we see that for every \(K \ge 1\) there exists a set \(P \subset Q_{0,\varepsilon }\) of measure \(\vert P\vert \le \frac{C\varepsilon ^{m+1}}{K}\), with \(C=C(m)\), such that for all \(y \in Q_{0,\varepsilon }{\backslash } P\) and \(l \in \{0,\ldots , m+1\}\) we have
Since we chose \({\hat{u}}\) and \({\hat{v}}\) with the absolute continuity properties described in Sect. 4.6 on \(\overline{Q_1(0)}\) it follows that for almost every \(x \in Q_{0,\varepsilon }\), with respect to the \(m+1\)-dimensional Lebesgue measure, all of the functions \({\hat{u}},{\hat{v}},\nabla {\hat{u}},\nabla {\hat{v}}\) are \({\mathcal {H}}^l\) almost everywhere defined on each of the l-dimensional faces of \(x + Q_{i,\varepsilon }\) for \(Q_{i,\varepsilon }\in {\mathcal {Q}}\) and \(l = 1,\ldots ,m+1\). Moreover, the gradients of \({\hat{u}}\) and \({\hat{v}}\) on any l-dimensional face of \(x + Q_{i,\varepsilon }\) coincide \({\mathcal {H}}^l\) almost everywhere with the tangential parts of \(\nabla {\hat{u}}\) and \(\nabla {\hat{v}}\) respectively. Thus we may choose \(x = a \in Q_{0,\varepsilon }\) such that these properties hold and, provided we choose K (depending on m) sufficiently large in (4.24), such that \(a_{m+1}\ge \frac{\varepsilon }{2}\) and such that we may apply (4.24) simultaneously for \(f(x) = \vert x_{m+1}\vert ^{\beta }{\tilde{f}}(x) \) with \({\tilde{f}}(x) =\vert {\hat{u}}(x)-{\hat{v}}(x)\vert ^2\) and \({\tilde{f}}(x) = \vert \nabla {\hat{u}}(x)\vert ^2+ \vert \nabla {\hat{v}}(x)\vert ^2 \) (where \(\nabla \) is the gradient on \({\mathbb {R}}^{m+1}\)). In particular, after discarding the integrals in (4.24) taken over any cube faces which do not intersect \({\mathbb {R}}^{m+1}_+\), we have
Now we begin the construction of w by defining a map on the one dimensional faces of every \(Q \times [0,\varepsilon ]\) where \(Q := (a + Q_{i,\varepsilon })\cap {\mathbb {R}}^{m+1}_+\) with \(Q_{i,\varepsilon } \in {\mathcal {Q}}\) and \(i_{m+1} \ge -1\). Let \(E_j\) denote a one dimensional face of Q parallel to the jth coordinate axis for \(j = 1,\ldots ,m+1\). Define \(w(x,0) = {\hat{u}}(x)\) on \(Q\times \{0\}\) and \(w(x,\varepsilon ) = {\hat{v}}(x)\) on \(Q\times \{\varepsilon \}\) and let \(w(x,s) = (1 - \frac{s}{\varepsilon }) {\hat{u}}(x) + \frac{s}{\varepsilon } {\hat{v}}(x)\) for \(x \in E_j\) and \(s \in [0,\varepsilon ]\). Since \({\hat{u}}({\mathbb {R}}^{m+1}_+) \subset N\) by definition, it follows that
for x in the 1-dimensional edges of Q and \(s \in [0,\varepsilon ]\).
We now estimate \(\sup _{E_j}\vert {\hat{u}} - {\hat{v}}\vert ^2\) using the Sobolev embedding theorem for \(W^{1,2}_{\beta }\) along the line segments \(E_j\), which one can deduce analogously to the case for \(W^{1,2}\) functions. Note that in our construction so far, we have discarded any edges \(E_j \subset \partial {\mathbb {R}}^{m+1}_+\). If \(E_j\), \(j = 1,\ldots ,m+1\), lies in a \(Q =(a + Q_{i,\varepsilon })\cap {\mathbb {R}}^{m+1}_+\) with \(i_{m+1} \ge -1\), with the exception of the case \( i_{m+1} = -1\), \(j = m+1\) and \(\beta \in (0,1)\), we calculate
If \(\beta > 0\), \( i_{m+1} = -1\) and \(j = m+1\) then we calculate
for any \(p \in (1,\frac{2}{1+\beta })\). The combination of (4.26), (4.27) and (4.28) with (4.25), applied with \(l=1\), yields
where \(q \in \{p,2\}\), for p fixed as above, depends on \(\beta \).
Next we bound the gradient of w on the product of the 1-dimensional edges of Q with \([0,\varepsilon ]\). Let \({\overline{\nabla }}\) denote the gradient on \(E_j \times [0,\varepsilon ]\). Recall that \({\hat{u}},{\hat{v}}\) are defined so that the tangential parts of their gradients \(\nabla {\hat{u}},\nabla {\hat{v}}\) on \({\mathbb {R}}^{m+1}_+\) coincide \({\mathcal {H}}^1\) almost everywhere with their gradients on the edges \(E_j\). It follows that
for x in any edge \(E_j\), \(j = 1,\ldots ,m+1\), of Q. Integrating over \(E_j \times [0,\varepsilon ]\) with respect to \(x_{m+1}^{\beta }\mathrm {d}x_j\mathrm {d}s\) yields
Consider again \(Q \times [0,\varepsilon ]\) for \(Q = (a+Q_{i,\varepsilon })\cap {\mathbb {R}}^{m+1}_+\) with \(Q_{i,\varepsilon }\in {\mathcal {Q}}\) and \(i_{m+1} \ge -1\). Recall that we are excluding cube faces in \(\partial {\mathbb {R}}^{m+1}_+\) from our construction. We use a slightly different procedure to extend w to higher dimensions depending on whether \(i_{m+1} = -1\) or \(i_{m+1} \ge 0\). Accordingly we introduce some temporary notation for two classes of \(F^l\) that we consider. Let \(F^l_{\bot }\) denote any l-dimensional face of any Q with no edges in the \(m+1\) direction and let \(F^l_{m+1}\) denote any face of any such Q with edges in the \(m+1\) direction. Suppose that \(l \ge 2\) and w is already defined with \(L^2\) gradient on every \(F^{l}_{\bot }\) and \(F^{l-1}_{\bot }\times [0,\varepsilon ]\), square \(x_{m+1}^{\beta }\mathrm {d}{\mathcal {H}}^{l}\)-integrable gradient on every \(F_{m+1}^{l}\) and square \(x_{m+1}^{\beta }\mathrm {d}{\mathcal {H}}^{l-1}\mathrm {d}s\)-integrable gradient on \(F_{m+1}^{l-1}\times [0,\varepsilon ]\). Furthermore, suppose that \(w(x,0) = {\hat{u}}(x)\) and \(w(x,\varepsilon ) = {\hat{v}}(x)\) for \(x \in F^{l}_{\bot }\) or \(x \in F^{l}_{m+1}\). These assumptions imply that w is defined \({\mathcal {H}}^l\) almost everywhere on all the l-dimensional faces of Q for \(l \ge 2\). Since \(\partial (F^l_{\bot } \times [0,\varepsilon ])\) and \(\partial ^+(F^l_{m+1} \times [0,\varepsilon ])\) are the union of such l-dimensional faces, w is defined \({\mathcal {H}}^l\) almost everywhere on these sets. If Q is such that \(i_{m+1} \ge 0\) then we do not distinguish between \(F^l_{\bot }\) and \(F^l_{m+1}\) and extend w to each \(F^l_{\bot } \times [0,\varepsilon ]\) and \(F^l_{m+1} \times [0,\varepsilon ]\) by homogeneous extension of degree zero with respect to \((y,\frac{\varepsilon }{2})\), where y is the centre point of \(F^l_{\bot }\) or \(F^l_{m+1}\). If \(i_{m+1} = -1\) then we extend w into \(F^l_{\bot } \times [0,\varepsilon ]\) using the same method. We extend w homogeneously of degree 0 from \(\partial ^+(F^l_{m+1}\times [0,\varepsilon ])\) into \(F^l_{m+1}\times [0,\varepsilon ]\) with respect to the point \(\left( y^+,\frac{\varepsilon }{2}\right) \), where y is the centre point of \(F^l_{m+1}\) and \(y^+ = y - (0,y_{m+1})\).
Now let \(F^l\) denote any l-dimensional face of any Q again. Since the tangential parts of the gradients \(\nabla {\hat{u}},\nabla {\hat{v}}\) on \({\mathbb {R}}^{m+1}\) coincide with the gradients of \({\hat{u}}\) and \({\hat{v}}\) on \(F^l\) for \({\mathcal {H}}^l\) almost every \(x \in F^l\), using the fact that \({\hat{u}}\) and \({\hat{v}}\) are homogeneous of degree zero, we calculate
where \({\overline{\nabla }}\) is the gradient on the set it is integrated over. From (4.31), we inductively deduce that for any \(l \in \{2,\ldots ,m+1\}\) we can extend w to each \(F^l\times [0,\varepsilon ]\) in \(Q\times [0,\varepsilon ]\) so that w has an \(L^2\) or \(x_{m+1}^{\beta }\mathrm {d}{\mathcal {H}}^l\mathrm {d}s\)-integrable gradient \({\overline{\nabla }}w\) on these faces. Moreover, \({\overline{\nabla }}w\) satisfies
So far, we have constructed a map \(w = w^{i,\varepsilon }\) on each cube and rectangle \(Q = (a+Q_{i,\varepsilon } )\cap {\mathbb {R}}^{m+1}_+\) such that \(Q_{i,\varepsilon } \in {\mathcal {Q}}\) with \(i_{m+1} \ge -1\). It follows from the construction that \(w^{(i,\varepsilon )} = w^{(j,\varepsilon )}\) \({\mathcal {H}}^{l+1}\)-almost everywhere on common faces \(F^l \times [0,\varepsilon ]\) of \((a + Q_{i,\varepsilon })\cap {\mathbb {R}}^{m+1}_+\) and \((a + Q_{j,\varepsilon })\cap {\mathbb {R}}^{m+1}_+\). Furthermore, for \(0<\varepsilon <{\frac{1}{8}}\) it follows that
We may therefore define \(w \in {W}^{1,2}_{\beta }(Q^+_{\frac{1}{4}}(0)\times [0,\varepsilon ];{\mathbb {R}}^n)\) by \(w\vert _{(a + Q_{i,\varepsilon })\cap {\mathbb {R}}^{m+1}_+}(x,s) = w^{(i,\varepsilon )}(x,s)\) for \(s \in [0,\varepsilon ]\). Since w is homogeneous of degree 0 on any l-dimensional face of any \(Q\times [0,\varepsilon ]\) with \(l \ge 3\), our inductive procedure preserves (4.29) for all (x, s) in \(Q^+_{\frac{1}{4}}(0)\times [0,\varepsilon ]\), with the possible exception of a set P of m-dimensional Hausdorff measure 0. It follows from (4.29) that for \((x,s) \in (Q^+_{\frac{1}{4}}(0)\times [0,\varepsilon ]){\backslash } P\) we have
where \(q \in \{2,p\}\) for some fixed \(p \in (1,\frac{2}{1+\beta })\). Moreover, we combine (4.30), (4.32) and (4.25) to see that
The definition of w as required now follows from combining (4.33) and (4.34) with (4.22) and (4.23). The absolute continuity properties, described in Sect. 4.6, of w, viewed as a function defined on a rectangle in polar coordinates, guarantee that for almost every \(\rho \in [\frac{1}{8}, \frac{1}{4}]\), w has square \(x_{m+1}^{\beta }\mathrm {d}{\mathcal {H}}^{m}\mathrm {d}s\)-integrable gradient on \(\partial ^+ B_{\rho }^+(0)\times [0,\varepsilon ]\) which coincides \({\mathcal {H}}^{m}\mathrm {d}s\) almost everywhere with the tangential part of \({\overline{\nabla }} w\). Using Fubini’s theorem and Chebychev’s inequality, applied to the map \(\rho \mapsto \int _{\partial ^+ B_{\rho }^+(0)\times [0,\varepsilon ]}x_{m+1}^{\beta }\vert {\overline{\nabla }}w\vert ^2\mathrm {d}{\mathcal {H}}^{m}\mathrm {d}s\), we may therefore choose \(\rho \in [\frac{1}{8}, \frac{1}{4}]\) such that w has square \(x_{m+1}^{\beta }\mathrm {d}{\mathcal {H}}^{m}\mathrm {d}s\)-integrable gradient on \(\partial ^+ B_{\rho }^+(0)\times [0,\varepsilon ]\) and satisfies
We define \({\tilde{w}}\) on \({\mathbb {S}}_+^m \times [0,\varepsilon ]\) by \({\tilde{w}}(\omega ,s) = {w}(\rho \omega ,s)\) and observe that this map has the required properties. \(\square \)
4.8 Comparison maps
With Lemma 4.16 in hand, we may now construct comparison maps for \(W^{1,2}_{\beta }\) functions which have values in N and are defined on half-balls centred in \(\partial {\mathbb {R}}^{m+1}_+\), provided the re-scaled energy is sufficiently small. We use the notation \({\overline{v}}_{B_{\rho }^+(y),\beta } = \left( \int _{B_{\rho }^+(y)}x_{m+1}^{\beta }\mathrm {d}x\right) ^{-1}\int _{B_{\rho }^+(y)}x_{m+1}^{\beta }v\mathrm {d}x\).
Lemma 4.17
There exists a \(\delta _0 = \delta _0(m,N,\beta )>0 \) such that the following holds. Let \(\varepsilon \in (0,1)\) and \(v \in {W}^{1,2}_{\beta }(B_{\rho }^+(y);N)\) with \(\rho ^{1 - m - \beta }\int _{B_{\rho }^+(y)}x_{m+1}^{\beta }\vert \nabla v \vert ^2\mathrm {d}x \le \delta _0^2\varepsilon ^{m+1 +\frac{\beta }{2}+\frac{\vert \beta \vert }{2}}\), where \(y \in \partial {\mathbb {R}}^{m+1}_+\). Then there is a \(\sigma \in (\frac{3\rho }{4},\rho )\) such that we can find a map \(w_{\varepsilon } \in {W}^{1,2}_{\beta }(B_{\rho }^+(y);N)\) which agrees with v in \(B_{\rho }^+(y){\backslash } {B_{\sigma }^+(y)}\) and such that
for a constant \(C = C(m,\beta )\).
Proof
We follow the proof of Corollary 1 in Section 2.7 of [39]. Throughout, C denotes a constant which depends on m and possibly \(\beta \) and we only distinguish different C when necessary. We also assume, without loss of generality, that \(\varepsilon \le \frac{1}{2}\).
Let \(\delta _0 > 0\) to be chosen as required and suppose the assumptions of the lemma hold for \(\delta _0\). As a consequence of Poincaré Inequality for the \(A_2\) weights \(\vert x_{m+1}\vert ^{\beta }\) [19], we have
As discussed in Sect. 4.6, we may work with a representative of v, which we do not relabel, such that \(v(B^+_{\rho }(y)) \subset N\). It follows that
for every \(x \in B^+_{\rho }(y)\). Integrating (4.37) over \(B^+_{\rho }(y)\) with respect to \(x_{m+1}^{\frac{\beta }{2} + \frac{\vert \beta \vert }{2}}\mathrm {d}x\) and then dividing by \(\int _{B^+_{\rho }(y)}x_{m+1}^{\frac{\beta }{2} + \frac{\vert \beta \vert }{2}}\mathrm {d}x\) we see that
Combining (4.38) with (4.36) we find
Hence, we may choose \(\lambda \in N\) such that
Using Chebychev’s inequality, we choose a \(C >0\) such that there exists \(\sigma \in (\frac{3\rho }{4},\rho )\) such that \({\hat{v}} \in {W}^{1,2}_{\beta }({\mathbb {S}}^{m}_+ ;N)\), where \({\hat{v}}(\omega ) = v(\sigma \omega +y)\), and such that
where D is the gradient on \({\mathbb {S}}^{m}_+\), and
We may therefore apply Lemma 4.16 to \({\hat{v}} \in {W}^{1,2}_{\beta }({\mathbb {S}}^{m}_+ ;N)\) and \(\lambda \). This yields a \(w_0{:}\,{\mathbb {S}}^m_+ \times [0,\varepsilon ] \rightarrow {\mathbb {R}}^n\) with \(w_0 = {\hat{v}}\) on \({\mathbb {S}}^m_+\times \{0\}\) and \(w_0 = \lambda \) on \({\mathbb {S}}^m_+\times \{\varepsilon \}\) in the sense of traces. Furthermore, (4.20) yields
where \({\overline{D}}\) is the gradient on \({\mathbb {S}}^m_+\times [0,\varepsilon ]\) and D is the gradient on \({\mathbb {S}}^m_+\). In addition, (4.21) implies that
for every \((\omega ,s) \in {\mathbb {S}}^m_+\times [0,\varepsilon ]\), where \(q \in (1,2]\) depends on \(\beta \). Henceforth we assume that \(\delta _0 \le 1\). Using (4.39) and (4.41) we deduce that
The combination of (4.43) with (4.40) and (4.44) yields
for every \((\omega ,s) \in {\mathbb {S}}^m_+\times [0,\varepsilon ]\).
Choose \(\delta _0\), depending on \(N,m,\beta \), such that \( C\delta _0\le {\hat{\alpha }}\) where C is the constant in (4.45) and \({\hat{\alpha }} > 0\) is sufficiently small to guarantee that the nearest point projection \(\pi _N\) onto N exists and has bounded derivatives in \(N_{{\hat{\alpha }}} = \{x \in {\mathbb {R}}^n{:}\,\text {dist}(x,N)\le {\hat{\alpha }}\}\). It then follows from (4.45) that we may apply \(\pi _N\) to \(w_0\). Let \(\omega \in {\mathbb {S}}^{m}_+\) satisfy \(\omega = \omega (x) = \frac{x-y}{\vert x-y\vert }\), \(r = \vert x - y\vert \) and define \(w \in {W}^{1,2}_{\beta }(B_{\rho }^+(y);N)\) by
Note that w agrees with v in \(B_{\rho }^+(y){\backslash } {B_{\sigma }^+(y)}\). We then readily calculate that w satisfies (4.35) as required. \(\square \)
4.9 Control of the mean squared oscillation
The Euler–Lagrange equations of \(E^{\beta }\) satisfy the structural conditions \(\vert \text {div}(x_{m+1}^{\beta }\nabla v)\vert \le c_0x_{m+1}^{\beta }\vert \nabla v\vert ^2\), together with the Neumann condition (4.2). For functions satisfying such conditions, if the re-scaled energy is sufficiently small it is possible to control their mean squared oscillation using the energy as follows.
Lemma 4.18
For every \(\delta >0\) and every \(c_0>0\) there exist two constants \(\varepsilon = \varepsilon ( m,n,\delta ,c_0)>0\) and \(\theta = \theta (m,n,\delta ,c_0) \in (0,\frac{1}{4}]\) such that the following holds. Let \(x_0 \in \partial {\mathbb {R}}^{m+1}_+\), \(R>0\) and \(B^+_{R}(x_0 ) \subset {\mathbb {R}}^{m+1}_+\). Suppose \(v \in {W}^{1,2}_{\beta }(B^+_R(x_0);{\mathbb {R}}^n)\) satisfies
for every \(\phi \in C^{\infty }_0(B_R(x_0);{\mathbb {R}}^n)\). If
then
Proof
The proof of the lemma is based on a blow-up procedure analogous to that of the proof of Lemma 3.5 in [29] for example.
Observe that the statement of the lemma is invariant under rescaling and translation by any point in \(\partial {\mathbb {R}}^{m+1}_+\); henceforth we assume \(R = 1\) and \(x_0 = 0\). Suppose, for a contradiction, that there exist \(\delta > 0\) and \(c_0>0\) such that the claim is false. Then for any \(\theta \in (0,\frac{1}{4}]\) there is a sequence of maps \((v_k)_{k\in {\mathbb {N}}}\), with \(v_k \in {W}^{1,2}_{\beta }(B^+_1(0);{\mathbb {R}}^n)\) for every k, such that
for every \(\phi \in C^{\infty }_0(B_{1}( 0);{\mathbb {R}}^n)\) and
but
Consider the normalised sequence \((w_k)_{k \in {\mathbb {N}}}\) defined by \(w_k = {\varepsilon _k^{-\frac{1}{2}}}( v_k - \overline{(v_k)}_{B^+_{\theta }(0),\beta })\). Then \(\nabla w_k = \varepsilon _k^{-\frac{1}{2}}\nabla v_k\) and thus
Furthermore, we deduce from (4.47) that
Using (4.48) and the Poincaré inequality for \(A_2\) weights, we deduce that the sequence \((w_k)_{k \in {\mathbb {N}}}\) is bounded in \(W^{1,2}_{\beta }(B^+_1(0);{\mathbb {R}}^n)\). Hence, the Compactness Lemma, Lemma 2.5, yields a subsequence \((w_{k_j})_{j \in {\mathbb {N}}}\) which converges weakly in \(W^{1,2}_{\beta }(B^+_1(0);{\mathbb {R}}^n)\) and strongly in \(L^2_{\beta }(B^+_1(0);{\mathbb {R}}^n)\) to a \(w \in W^{1,2}_{\beta }(B^+_1(0);{\mathbb {R}}^n)\).
In view of (4.46) and (4.48) we calculate
for every \(\phi \in C^{\infty }_0(B_1(0);{\mathbb {R}}^n)\). Since \(w_{k_j} \rightharpoonup w\) in \(W^{1,2}_{\beta }(B^+_1(0);{\mathbb {R}}^n)\), it follows that
for every \(\phi \in C_0^{\infty }(B_1(0);{\mathbb {R}}^n)\). Hence w is a weak solution of the linear Neumann-type problem (2.11) and, in particular, satisfies (2.12) from Lemma 2.7 in \(B^+_1(0)\).
We also conclude, using the Compactness Lemma to take limits in (4.48) and (4.49), that
and
respectively. Now, since \({\overline{w}}_{B^+_{\theta }(0),\beta } =0\), the Poincaré inequality yields
We apply Lemma 2.7 to w with \(\theta \le \frac{1}{4}\) (so that \(2\theta \le \frac{1}{2}\)). This gives a positive constant C (independent of \(\theta \)) and a \(\gamma \in (0,1)\) such that
Combining (4.51) and (4.52) we see that
This holds for all fixed \(\theta \in (0,\frac{1}{4}]\) and we choose \(\theta < \frac{1}{2}\left( \frac{\delta }{C}\right) ^{\frac{1}{2\gamma }}\) so that (4.53) contradicts (4.50). \(\square \)
Remark 4.19
We could have used Lemma 2.8 in place of Lemma 2.7 to the same effect. In using the latter lemma, we observe that Hölder continuity of solutions to the linear Neumann-type problem (2.11) is sufficient to obtain energy decay and consequently Hölder continuity of minimisers of \(E^{\beta }\) relative to \({\mathcal {O}}\); we do not need higher regularity for the linear problem at this point.
4.10 Energy decay
We combine our construction of comparison maps in Sect. 4.8 with the improved control of the mean squared oscillation obtained in Sect. 4.9 in order to show that the re-scaled energy decays faster than implied by the boundary monotonicity formula, Lemma 4.5.
Lemma 4.20
Let \(v \in {\dot{W}}_{\beta }^{1,2}({\mathbb {R}}^{m+1}_+;N)\) be a minimiser of \(E^{\beta }\) relative to \({\mathcal {O}}\). Suppose \(B^+_R(x_0)\) is a half-ball with \(R \le 1\) and \(\overline{\partial ^0B^+_{R}(x_0)}\subset {\mathcal {O}}\). There exist \(\varepsilon _0 = \varepsilon _0(m,N,\beta )>0\) and \(\theta _0=\theta _0(m,N,\beta ) \in (0, \frac{1}{4})\) such that if
then
for every \(B^+_r(y) \in {\mathcal {B}}^+(x_0,R, \frac{R}{2})\).
Proof
Let \(B^+_{\rho }(y) \subset B^+_r(y) \in {\mathcal {B}}^+(x_0,R, \frac{R}{2})\). Then \(\rho \le r \le \frac{R}{2}\), \(y \in {\mathcal {O}}\) and \(\vert x_0-y\vert <\frac{R}{2}\). Suppose v satisfies \(R^{1-m - \beta }\int _{B^+_R(x_0)}x_{m+1}^{\beta }\vert \nabla v\vert ^2\mathrm {d}x \le \varepsilon _0\) for \(\varepsilon _0 >0\) to be chosen. Then for any \(\rho \in (0,r]\) the monotonicity formula, Lemma 4.5, yields
We apply Lemma 4.17 on \(B^+_\rho (y)\subset B^+_r(y)\), with \(\rho \le r\) to be chosen later. This gives a \(\delta _0\) such that for any \(\varepsilon \in (0,1)\), if
then there is a \(\sigma \in (\frac{3\rho }{4},\rho )\) such that we can find a \(w_{\varepsilon } \in {W}^{1,2}_{\beta }(B_{\rho }^+(y);N)\) which agrees with v in \(B_{\rho }^+(y){\backslash } {B_{\sigma }^+(y)}\) and satisfies
Assuming (4.55) and consequently (4.56) hold, we make use of the comparison property of w. Since \(v = w\) in \(B_{\rho }^+(y){\backslash } {B_{\sigma }^+(y)}\) we may extend w to an element of \({\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N)\) by requiring \(w = v\) on \({\mathbb {R}}^{m+1}_+{\backslash } B_{\rho }^+(y)\). As v is a minimiser of \(E^{\beta }\) relative to \({\mathcal {O}}\), we deduce that \(\int _{B_{\sigma }^+(y)}x_{m+1}^{\beta }\vert \nabla v \vert ^2\mathrm {d}x \le \int _{B_{\sigma }^+(y)}x_{m+1}^{\beta }\vert \nabla w \vert ^2\mathrm {d}x\). Combining this fact with the monotonicity formula, Lemma 4.5, and (4.56) gives
Fix \(\varepsilon = \min \{\frac{1}{4},\frac{1}{4C}\}\), where C is the constant in (4.57) and let \(\varepsilon _0 \le \frac{1}{C} \delta _0^2\varepsilon ^{1+m + \frac{\beta }{2}+ \frac{\vert \beta \vert }{2}}\) where C is the constant from (4.54). It follows from (4.54) that (4.55) is satisfied and hence, substituting this \(\varepsilon \) into (4.57), we have
for a constant \({\hat{C}}\) and any \(\rho \le r \le \frac{R}{2}\). Observe that
for every \(\phi \in C^{\infty }_0(B_r(y);{\mathbb {R}}^n)\) on every \(B_r^+(y) \subset B_R^+(x_0)\) where \(c_0 = c_0(m,N)\). Hence, we may apply Lemma 4.18 for \(\delta = \min \{\frac{1}{4}, \frac{1}{4{\hat{C}}}\}\) and \( c_0\) as above to obtain a corresponding \(\varepsilon _1>0\) and \(\theta _1 \in (0,\frac{1}{4}]\) such that if \( r^{1-m-\beta }\int _{B^+_{r}(y)}x_{m+1}^{\beta }\vert \nabla v\vert ^2\mathrm {d}x\le \varepsilon _1\) then
Now choose \(\varepsilon _0 = \frac{1}{C}\min \{\delta _0^2 {\varepsilon }^{1+m + \frac{\beta }{2}+ \frac{\vert \beta \vert }{2}}, \varepsilon _1\}\) where C is the constant from (4.54). It follows that (4.58) and (4.59) hold on any \(B^+_{\rho }(y)\subset B^+_{r}(y) \in {\mathcal {B}}^+(x_0,R,\frac{R}{2})\). Thus we may apply (4.58) with \(\rho = \theta _1 r\). In turn, assuming this choice of \(\rho \), we combine (4.58) with the monotonicity formula and (4.59) to see that
Hence the lemma is proved with the above choice of \(\varepsilon _0\) and \(\theta _0 = \frac{3 \theta _1}{4}\). \(\square \)
4.11 \(\varepsilon \)-Regularity as far as Hölder continuity
The culmination of the results in this section so far lead to the following \(\varepsilon \)-regularity theorem for minimisers of \(E^{\beta }\) relative to \({\mathcal {O}}\) which establishes the first part of Theorem 4.1.
Theorem 4.21
If \(m \ge 3\), let \(\beta \in (-1,1)\) and if \(m =2\) let \(\beta \in (-3^{-1},1)\). Let \(v \in {\dot{W}}_{\beta }^{1,2}({\mathbb {R}}^{m+1}_+;N)\) be a minimiser of \(E^{\beta }\) relative to \({\mathcal {O}}\). Suppose \(B^+_R(x_0)\) satisfies \(R \le 1\) and \(\overline{\partial ^0B^+_{R}(x_0)}\subset {\mathcal {O}}\). There exists an \(\varepsilon = \varepsilon (m,N,\beta )>0\) and a \( \theta =\theta (m,N,\beta ) \in (0,1)\) such that if
then \(v \in C^{0,\gamma }(\overline{B^+_{\theta R}(x_0)};N)\) for some \(\gamma = \gamma (m,N,\beta ) \in (0,1)\). In particular,
for every \(x_1,x_2 \in B^+_{\theta R}(x_0)\) and a constant \(C = C(m,N,\beta )\).
Proof
Throughout the proof we adopt the convention that all constants depend only on m, N and \(\beta \) unless stated otherwise. We reinforce this dependence where appropriate.
Let v be a minimiser of \(E^{\beta }\) relative to \({\mathcal {O}}\) with \(R^{1-m - \beta }\int _{B^+_R(x_0)}x_{m+1}^{\beta }\vert \nabla v\vert ^2\mathrm {d}x \le \varepsilon \) and set \(\varepsilon = \min \{\varepsilon _0,\varepsilon _1\}\), where \(\varepsilon _0\) is the number from Lemma 4.20 and \(\varepsilon _1\) is the number from Lemma 4.9.
Observe that the function \( {\tilde{r}} \mapsto {\tilde{r}}^{1 - m -\beta }\int _{B^+_{{\tilde{r}}}(z)} x_{m+1}^{\beta }\vert \nabla v \vert ^2\mathrm {d}x\) is non-decreasing on \((0,\frac{R}{2}]\) by the monotonicity formula, Lemma 4.5. Furthermore, the choice of \(\varepsilon \) allows us to apply Lemma 4.20. We apply this lemma, together with Lemma 8.23 of [15] to deduce that on every \(B^+_{{\tilde{r}}}(z) \in {\mathcal {B}}^+(x_0,R, \frac{R}{2})\) we have
for a constant C and a \(\gamma _0 \in (0,1)\) which depend only on \(m,N,\beta \).
Our choice of \(\varepsilon \) also permits the application of Lemma 4.9; this lemma implies that for any \(B_r(y) \in {\mathcal {B}}_{\theta _1}(x_0,R,\frac{R}{3})\), with \(\theta _1\ge 2\) given by the lemma, and any \(0<\rho \le r\) we have
for some \(\gamma _1\in (0,1)\). Since \(\theta _1 \ge 2\), for any \(B_r(y) \in {\mathcal {B}}_{\theta _1}(x_0,R,\frac{R}{3})\) we have the inclusions
where \(y^+ = y - (0,y_{m+1})\). It follows, applying (4.6), that
where C depends on \(m,N,\beta \) and \(\theta _1\) and thus only on \(m,N,\beta \). We combine (4.62), applied with \(r = \frac{y_{m+1}}{\theta _1}\), with (4.63), (4.64) and (4.61), applied on \(B^+_{{\tilde{r}}}(z)\) with \({\tilde{r}} =\left( \frac{\theta _1+1}{\theta _1}\right) y_{m+1}\) and \(z = y^+\). It follows, after defining \({\hat{\gamma }} = \min \{\gamma _0,\gamma _1\}\), that
This holds for any \(B_{\rho }(y) \in {\mathcal {B}}_{\theta _1}(x_0,R,\frac{R}{3} )\). Since (4.61) holds on every \(B^+_{{\tilde{r}}}(z) \in {\mathcal {B}}^+(x_0,R,\frac{R}{2})\) for \(\gamma _0\), it holds on every \(B^+_{{\tilde{r}}}(z) \in {\mathcal {B}}^+(x_0,R, \frac{R}{3})\) for \({\hat{\gamma }}\). We deduce from (4.61) and (4.65) that the hypothesis of Lemma 4.8 are satisfied; applying this lemma concludes the proof. \(\square \)
Remark 4.22
Once we know that a minimiser of \(E^{\beta }\) relative to \({\mathcal {O}}\) is Hölder continuous in some \(B^+_R(x_0)\) with \(\overline{\partial ^0 B^+_R(x_0)}\subset {\mathcal {O}}\), known theory for harmonic maps, see Lemma 4.13 which is from [38], implies that v is smooth in \(B^+_R(x_0)\). However, this theory does not imply v is smooth up to \(\partial ^0 B^+_R(x_0)\); henceforth our goal is essentially to prove this fact.
4.12 An \(L^{\infty }\) bound for the gradient
The first step in our proof of higher partial regularity of locally minimising fractional harmonic maps consists of establishing an \(L^{\infty }\) bound for the gradients to solutions of systems of semi-linear equations with growth conditions satisfied by minimisers v of \(E^{\beta }\) relative to \({\mathcal {O}}\) and their derivatives \(D^{\alpha '}v\) where \(\alpha '\in {\mathbb {N}}^{m+1}_0\) is a multi-index with \(\alpha '_{m+1}=0\). The method of proof is that of harmonic replacement; compare the growth of the average Dirichlet energy of solutions of the semi-linear equations with that of solutions to the linearised system. We follow [38] for example. The monotonicity formulas established in Lemmas 2.8 and 2.17 are a key ingredient of the proof.
We will use the notation \(\vert {\varOmega }\vert _{\beta }= \int _{{\varOmega }}\vert x_{m+1}\vert ^{\beta }\mathrm {d}x\) and \(\vert {\varOmega }\vert = \int _{{\varOmega }} \mathrm {d}x\) for \({\varOmega }\subset {\mathbb {R}}^{m+1}\).
Lemma 4.23
Suppose \(v \in {W}^{1,2}_{\beta }(B^+_R(x_0);{\mathbb {R}}^n) \cap C^{0,\gamma }(\overline{B^+_R(x_0)};{\mathbb {R}}^n)\) where \(B^+_R(x_0)\) is a half-ball with \(x_0 \in \partial {\mathbb {R}}^{m+1}_+\), \(R \le 1\) and \(\gamma \in (0,1)\). Suppose v satisfies
for every \(\psi \in C_0^{\infty }(B_R(x_0);{\mathbb {R}}^n)\), where \(G{:}\,{\mathbb {R}}^m\times {\mathbb {R}}^{(m+1)n}\) is measurable and \(\vert G(x,q) \vert \le C_1\vert q \vert ^2+ C_2\) for a positive \(C_1\le C^*\) and non-negative \(C_2 \le C^*\) for some \(C^*>0\). Then \(\nabla v \in L^{\infty }(B^+_{\frac{ R}{3}}(x_0);{\mathbb {R}}^{(m+1) n})\) and, in particular, we have
where \(C_3 = C_3(m,N,\beta , C^*)\), \(C_4 = C_4(m,N,\beta ,C^*)\) and \({\tilde{C}}_2 = C_2^{\frac{1}{2}}+C_2 \). In particular, if \(C_2 = 0\) then \({\tilde{C}}_2 = 0\).
Proof
Without relabelling, we reflect v evenly across the hyperplane \(\partial {\mathbb {R}}^{m+1}_+\). We observe that \(v \in C^{0,\gamma }(\overline{B_{ R}(x_0)};{\mathbb {R}}^n)\cap W^{1,2}_{\beta }(B_R(x_0);{\mathbb {R}}^n)\) is a weak solution of \(\text {div}(\vert x_{m+1}\vert ^{\beta } \nabla v) + \vert x_{m+1}\vert ^{\beta } {\tilde{G}} = 0\) in \(B_R(x_0)\), where \({\tilde{G}}\) is measurable and \(\vert {\tilde{G}}(x,q)\vert \le C_1\vert q\vert ^2+ C_2\). We will derive estimates on classes of \(B_{\rho }(y)\) with \(y \in B_\frac{R}{2}(x_0)\) and \(\rho \le \frac{R}{2}\). We focus initially on an estimate for the average energy on \(B_{\frac{\rho }{2}}(y)\) in terms of that on \(B_{\rho }(y)\). Since v is even with respect to \(\partial {\mathbb {R}}^{m+1}_+\) we only need to consider \(B_{\rho }(y)\) with \(y_{m+1}\ge 0\). We consider two cases, \(y_{m+1} = 0\) and \(y_{m+1}>0\).
Suppose \(B_{\rho }(y)\) is such that \(B^+_{\rho }(y) \in {\mathcal {B}}^+(x_0, R, \frac{ R}{2} )\). An application of Minkowski’s inequality, for maps in \(L^2_{\beta }(B_{\frac{\rho }{2}}(y);{\mathbb {R}}^{n(m+1)})\), yields
for any \(w \in W^{1,2}_{\beta }(B_{\rho }(y);{\mathbb {R}}^n)\). Let \(w \in W^{1,2}_{\beta }(B_{\rho }(y);{\mathbb {R}}^n)\) be the weak solution of \(\text {div} (\vert x_{m+1}\vert ^{\beta }\nabla w) = 0\) in \( B_{\rho }(y)\) with \(w = v\) on \(\partial B_{\rho }(y) \), given by Lemma 2.18. Then w is smooth in \(B_{\rho }(y) {\backslash } \partial {\mathbb {R}}^{m+1}_+\) and continuous in \(\overline{B_{\rho }(y) }\). Furthermore, since v is symmetric with respect to \(\partial {\mathbb {R}}^{m+1}_+\), it follows from Lemma 2.19 that w is symmetric with respect to \(\partial {\mathbb {R}}^{m+1}_+\) and, crucially, we are now free to apply Lemma 2.8 to w for every \(\beta \in (-1,1)\).
As \(w - v \in C(\overline{B_{\rho }(y)};{\mathbb {R}}^n) \cap W^{1,2}_{\beta ,0}(B_{\rho }(y);{\mathbb {R}}^n)\) and v satisfies \(\text {div}(\vert x_{m+1}\vert ^{\beta } \nabla v) + \vert x_{m+1}\vert ^{\beta }{\tilde{G}}= 0\) and w satisfies \(\text {div} (\vert x_{m+1}\vert ^{\beta }\nabla w) = 0\) weakly in \( B_{\rho }(y)\), we calculate
The Hölder continuity of v, together with the weak maximum and minimum principles given by Lemma 2.18 imply
Next we use the monotonicity and minimising properties of w to scale its averaged energy. An application of Lemma 2.8, followed by an application of Lemma 2.18 yields
Combining (4.66)–(4.69) we see that
Define \({\tilde{C}}_1 = (C_1+C_2)^{\frac{1}{2}} + C_1\) and \({\tilde{C}}_2 = C_2^{\frac{1}{2}}+C_2\). We square both sides of the preceding inequality, using Young’s inequality (\(ab \le \frac{a^2}{2}+ \frac{b^2}{2}\) for \(a,b \ge 0\)) and the fact that \(\gamma \in (0,1)\) and \(\rho \le R \le 1\) to see that
This holds on every \(B_{\rho }(y)\) with \(B^+_{\rho }(y) \in {\mathcal {B}}^+(x_0, R, \frac{ R}{2})\).
Now we iterate this estimate on concentric balls. Consider \(B_{\frac{ R}{2}}(y)\) with \(B^+_{\frac{ R}{2}}(y)\in {\mathcal {B}}^+(x_0, R, \frac{ R}{2})\). Let \(\rho _k = 2^{-k}\frac{ R}{2}\) for \(k \in {\mathbb {N}}_0\). First notice that \(\rho _k^{\frac{\gamma }{2}} \le 2^{-\frac{k\gamma }{2}}\). Hence
where \({\tilde{C}}\) depends on \(m,N,\beta \) and \( {C}^*\). It follows from (4.70) that, for every \(k \ge 1\), we have
Now we consider estimates on \(B_{\rho }(y)\) with \( B_{\frac{y_{m+1}}{2 }}(y) \in {\mathcal {B}} (x_0, R, \frac{ R}{3})\); on this class of ball we have
where \(y^+ = y - (0,y_{m+1})\). Let \(w \in W^{1,2}_{\beta }(B_{\rho }(y);{\mathbb {R}}^n)\) be the weak solution of \(\text {div} (\vert x_{m+1}\vert ^{\beta }\nabla w) = 0\) in \( B_{\rho }(y)\) with \(w = v\) on \(\partial B_{\rho }(y) \), given by Lemma 2.18 and suppose \(\theta \ge 2\) is such that \(y_{m+1} \ge \theta \rho \). Then Lemma 2.17 yields
Hence, repeating (4.66)–(4.69) but with \(\vert B_\frac{\rho }{2}(y)\vert _{ \beta }\) replaced by \(\vert B_\frac{\rho }{2}(y)\vert \), we find
We square both sides of this inequality analogously to (4.70), noting that \(\theta \ge 2\) so \(\frac{1}{\theta -1}\le 1\), \(\gamma \in (0,1)\) and \(\rho \le R \le 1\), to see that
This holds on every \(B_{\rho }(y)\) with \( B_{\frac{y_{m+1}}{2 }}(y) \in {\mathcal {B}} (x_0, R, \frac{ R}{3})\) and \(y_{m+1}\ge \theta \rho \).
We iterate this estimate on concentric balls. Consider \( B_{\frac{y_{m+1}}{2 }}(y) \in {\mathcal {B}} (x_0, R, \frac{ R}{3})\). Let \(\rho _k = 2^{-k}\frac{y_{m+1}}{2}\) for \(k \in {\mathbb {N}}_0\) and observe that \(y_{m+1} \ge 2^{k+1}\rho _k\) and \(\rho _k^{\frac{\gamma }{2}} \le 2^{-\frac{k\gamma }{2}}\). Observe
where \({\hat{C}}\) depends on \(m,N,\beta \) and \(C^*\). It follows from (4.73) that, for every \(k \ge 1\), we have
Now fix \(y \in B^+_{\frac{R}{3}}(x_0)\) which implies \( B_{\frac{y_{m+1}}{2 }}(y) \in {\mathcal {B}} (x_0, R, \frac{ R}{3})\). We divide (4.74) by \(y_{m+1}^{\beta }\), let \(y^+ = y - (0,y_{m+1})\) and combine (4.71), (4.72) and (4.74) to see that
An application of Lebesgue’s differentiation theorem concludes the proof. \(\square \)
Remark 4.24
A consequence of the preceding lemma is that Hölder continuous weak solutions of \(\text {div}( x_{m+1}^{\beta }\nabla v)+ x_{m+1}^{\beta }G =0\) in \(B^+_R(x_0)\) and \(x_{m+1}^{\beta }\partial _{m+1}v = 0\) in \(\partial ^0B^+_R(x_0)\), with G satisfying the assumptions of the lemma on \(B^+_R(x_0)\), are actually Lipschitz continuous on \(B^+_{\frac{R}{3}}(x_0)\).
4.13 Existence of higher order derivatives
The existence of higher order derivatives of minimisers of \(E^{\beta }\) relative to \({\mathcal {O}}\) in directions tangential to \(\partial {\mathbb {R}}^{m+1}_+\) follows using the usual method of difference quotients.
Lemma 4.25
Fix \(l \in {\mathbb {N}}_0\). Suppose v is a minimiser of \(E^{\beta }\) relative to \({\mathcal {O}}\) and let \(B^+_R(x_0)\) be a half-ball with \(\overline{\partial ^0B^+_R(x_0)}\subset {\mathcal {O}}\). Suppose further that for every multi-index \(\alpha ' \in {\mathbb {N}}^{m+1}_0\) with \(\alpha '_{m+1} = 0\) and \(\vert \alpha '\vert \le l\) we have \(D^{\alpha '}v \in C^{0,\gamma }(\overline{B^+_R(x_0)};{\mathbb {R}}^n) \cap {W}^{1,2}_{\beta }(B^+_R(x_0);{\mathbb {R}}^n)\) for some \(\gamma \in (0,1)\) and \(\nabla D^{\alpha '}v \in L^{\infty }(B^+_R(x_0);{\mathbb {R}}^{(m+1)n})\). Then for \(\alpha ' \in {\mathbb {N}}^{m+1}_0\) with \(\alpha '_{m+1} = 0\) and \(\vert \alpha '\vert = l\) and \(i = 1,\ldots ,m\), the weak derivative \(\nabla \partial _i D^{\alpha '}v\) exists and \( \nabla \partial _i D^{\alpha '}v \in L^2_{\beta }(B^+_{\frac{R}{2}}(x_0);{\mathbb {R}}^{(m+1)n})\).
Proof
Without relabelling, we extend A to a smooth section of \( T^{*}{\mathbb {R}}^n\otimes T^{*}{\mathbb {R}}^n\otimes T{\mathbb {R}}^n\). Fix \(\alpha ' \in {\mathbb {N}}^{m+1}_0\) with \(\alpha '_{m+1} = 0\) and \(\vert \alpha '\vert = l\). Since v is a minimiser of \(E^{\beta }\) relative to \({\mathcal {O}}\), the regularity assumptions on v and \(D^{\alpha '}v\) imply we may integrate by parts l times in (4.1); for any \(\phi \in C^{\infty }_0(B_R(x_0);{\mathbb {R}}^n)\) we have
Let \(\eta \in C^{\infty }_0(B_{\frac{3R}{4}}(x_0))\) be a smooth cutoff function such that \(\eta \equiv 1\) in \(B_\frac{R}{2}(x_0)\), \(1 \ge \eta \ge 0\) in \(B_\frac{3R}{4}(x_0){\backslash } B_{\frac{R}{2}}(x_0)\) and \(\vert \nabla \eta \vert \le \frac{C}{R}\). Furthermore, let \({\varDelta }_i^hD^{\alpha '}v = h^{-1}(D^{\alpha '}v(x+he_i) - D^{\alpha '}v(x))\) be the difference quotient of \(D^{\alpha '}v\) and assume \(\vert h\vert <\frac{R}{4}\). Note that, by approximation, \(w = - {\varDelta }_i^{-h}(\eta ^2 {\varDelta }_i^{h} D^{\alpha '}v)\) is an admissible test function for (4.75). We substitute w into (4.75) and apply ‘integration by parts’ for difference quotients to see that
We now use Young’s inequality, \(ab \le \frac{a^2}{\delta 2} + \delta \frac{b^2}{2}\) for \(a,b \ge 0\) and \(\delta >0\), to move all of the terms involving \( {\varDelta }_i^{h}\nabla D^{\alpha '} v\) on the right hand side of (4.76) to the left hand side. We calculate
We need to estimate the term involving \({\varDelta }_i^hD^{\alpha '}(A(v)(\nabla v,\nabla v))\) in a similar fashion. An application of the Mean Value Theorem, noting we are working on \( B^+_{\frac{3R}{4}}(x_0)\) and \(\vert h\vert <\frac{R}{4}\), implies
where \(C_1,C_2\) depend on \(m,N,\beta \) and \(\vert \vert \nabla D^{{\tilde{\alpha }}'}v\vert \vert _{L^{\infty }(B^+_R(x_0);{\mathbb {R}}^{(m+1)n})}\) where \( {\tilde{\alpha }}'\in {\mathbb {N}}^{m+1}_0\) with \( \vert {\tilde{\alpha }}'\vert \le \vert \alpha '\vert \) and \({\tilde{\alpha }}'_{m+1}=0\). Hence, using Young’s inequality again, we deduce
where \(C_3\) depends on \(m,N,\beta \) and \(\vert \vert \nabla D^{{\tilde{\alpha }}'}v\vert \vert _{L^{\infty }(B^+_R(x_0);{\mathbb {R}}^{(m+1)n})}\) where \( {\tilde{\alpha }}'\in {\mathbb {N}}^{m+1}_0\) with \( \vert {\tilde{\alpha }}'\vert \le \vert \alpha '\vert \) and \({\tilde{\alpha }}'_{m+1}=0\). Choosing \(\delta \) sufficiently small in (4.77) and (4.78) we combine these inequalities with (4.76). Since \(\vert \nabla \eta \vert \le \frac{C}{R}\) and \(\eta \equiv 1\) in \(B^+_{\frac{R}{2}}(x_0)\), we see that
where \(C_4\) depends on \(m,N,\beta \) and \(\vert \vert \nabla D^{{\tilde{\alpha }}'}v\vert \vert _{L^{\infty }(B^+_R(x_0);{\mathbb {R}}^{(m+1)n})}\) where \( {\tilde{\alpha }}'\in {\mathbb {N}}^{m+1}_0\) with \( \vert {\tilde{\alpha }}'\vert \le \vert \alpha '\vert \) and \({\tilde{\alpha }}_{m+1}' = 0\). This bound is independent of h with \(\vert h\vert <\frac{R}{4}\). Hence by Lemma 2.11 we conclude that the weak derivative \(\nabla \partial _i D^{\alpha '}v\) exists and satisfies the above inequality with \(\nabla \partial _i D^{\alpha '}v\) in place of \( {\varDelta }_i^{h} \nabla D^{\alpha '}v \). This concludes the proof. \(\square \)
4.14 Caccioppoli-type inequality
Here we show that the derivatives of minimisers of \(E^{\beta }\) relative to \({\mathcal {O}}\) in directions tangential to \(\partial {\mathbb {R}}^{m+1}_+\) all satisfy essentially the same Caccioppoli-type inequality.
Lemma 4.26
Fix \(l \in {\mathbb {N}}_0\), let \(v \in {\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N)\) be a minimiser of \(E^{\beta }\) relative to \({\mathcal {O}}\) and let \(B^+_R(x_0)\) be a half-ball with \(R \le 1\) and \(\overline{\partial ^0B^+_R(x_0)}\subset {\mathcal {O}}\). Suppose that for every multi-index \(\alpha ' \in {\mathbb {N}}^{m+1}_0\) with \(\alpha '_{m+1} = 0\) and \(\vert \alpha '\vert \le l\) we have \(D^{\alpha '}v \in C^{0,\gamma }(\overline{B^+_R(x_0)};{\mathbb {R}}^n) \cap {W}^{1,2}_{\beta }(B^+_R(x_0);{\mathbb {R}}^n)\) for some \(\gamma \in (0,1)\) and \(\nabla D^{\alpha '}v \in L^{\infty }(B^+_R(x_0);{\mathbb {R}}^{(m+1)n})\). Suppose further that for \(\alpha ' \in {\mathbb {N}}^{m+1}_0\) with \(\alpha '_{m+1} = 0\) and \(\vert \alpha '\vert = l\) and some \(i \in \{ 1,\ldots ,m\}\), we have \( \nabla \partial _i D^{\alpha '}v \in L^2_{\beta }(B^+_{R}(x_0);{\mathbb {R}}^{(m+1)n})\). Let \(B_\rho (y)\subset B_R(x_0)\) with \(y_{m+1} \ge 0\). For each \(\alpha '\) with \(\alpha '_{m+1}=0\) and \(\vert \alpha '\vert = l\) there are constants \(C=C(m,N,\beta )\) and \(C_1,C_2\) which depend on \(m,N,\beta \) and are comprised of polynomial functions, with no constant terms, of \(\vert \vert \nabla D^{{\tilde{\alpha }}'}v\vert \vert _{L^{\infty }(B_\rho (y)\cap {\mathbb {R}}^{m+1}_+;{\mathbb {R}}^{(m+1)n})}\), where \( {\tilde{\alpha }}'\in {\mathbb {N}}^{m+1}_0\) with \(\vert {\tilde{\alpha }}'\vert \le \vert \alpha '\vert =l\) and \({\tilde{\alpha }}'_{m+1}=0\), such that
for any \(\lambda \in {\mathbb {R}}^n\).
Proof
Fix \(\alpha '\) with \(\alpha '_{m+1}=0\) and \(\vert \alpha '\vert = l\). Integrating by parts \(l+1\) times in (4.1) shows that for every \(\psi \in C_0^{\infty }(B_R(x_0);{\mathbb {R}}^n)\), we have
Now, by approximation, we may choose \(\psi = \eta ^2 (\partial _iD^{\alpha '}v - \lambda )\) where \(\lambda \in {\mathbb {R}}^n\) is a constant vector and \(\eta \in C^{\infty }_0(B_{\rho }(y))\) is a cutoff function with \(\eta \equiv 1\) in \(B_{\frac{\rho }{2}}(y)\), \(0 \le \eta \le 1\), and \(\vert \nabla \eta \vert \le \frac{C}{\rho }\). We calculate
where \(C=C(m,N,\beta )\) and \({\tilde{C}}_1,{\tilde{C}}_2\) depend on \(m,N,\beta \) and are comprised of polynomial functions, with no constant terms, of \(\vert \vert \nabla D^{{\tilde{\alpha }}'}v\vert \vert _{L^{\infty }(B_{\rho }(y)\cap {\mathbb {R}}^{m+1}_+;{\mathbb {R}}^{(m+1)n})}\) where \({\tilde{\alpha }}'\in {\mathbb {N}}^{m+1}_0\) with \(\vert {\tilde{\alpha }}'\vert \le l\) and \({\tilde{\alpha }}_{m+1}' = 0\). We apply Young’s inequality, \(ab\le \frac{\delta a^2}{2} + \frac{b^2}{\delta 2}\) for \(a,b\ge 0\) and \(\delta >0\), to each term on the right hand side of (4.80). We first apply this inequality to the term corresponding to C, choosing \(\delta \) sufficiently small depending on C and hence only on \(m,N,\beta \) and recalling \(\vert \nabla \eta \vert \le \frac{C}{\rho }\) to see that
where \({\tilde{C}}_1,{\tilde{C}}_2\) are possibly different from before but have the same structure and dependence as the constants in (4.80). Applying Young’s inequality again, now to the terms in (4.81) corresponding to \({\tilde{C}}_1,{\tilde{C}}_2\) concludes the proof. \(\square \)
4.15 Control of the mean squared oscillation of the derivatives on the boundary
We prove an analogue of Lemma 4.18 for the derivatives of minimisers of \(E^{\beta }\) relative to \({\mathcal {O}}\).
Lemma 4.27
Fix \(l \in {\mathbb {N}}_0\). For every \(\delta > 0\) there exist numbers \(\varepsilon >0\), \(\tau \in (0,1)\) and \(\theta \in (0,\frac{1}{4}]\) such that the following holds. Suppose \(v \in {\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N)\) is a minimiser of \(E^{\beta }\) relative to \({\mathcal {O}}\) with \(D^{\alpha '}v \in C^{0,\gamma }(\overline{B^+_R(x_0)};{\mathbb {R}}^n) \cap {W}^{1,2}_{\beta }(B^+_R(x_0);{\mathbb {R}}^n)\) for a \(\gamma \in (0,1)\) and \(\nabla D^{\alpha '}v \in L^{\infty }(B^+_R(x_0);{\mathbb {R}}^{(m+1)n})\) for every multi-index \(\alpha ' \in {\mathbb {N}}^{m+1}_0\) with \(\alpha '_{m+1} = 0\) and \(\vert \alpha '\vert \le l\), where \(B^+_{R}(x_0 ) \subset {\mathbb {R}}^{m+1}_+\) satisfies \(R \le 1\) and \(\overline{\partial ^0 B^+_R(x_0)} \subset {\mathcal {O}}\). Suppose further that for \(\alpha ' \in {\mathbb {N}}^{m+1}_0\) with \(\alpha '_{m+1} = 0\) and \(\vert \alpha '\vert = l\) and some \(i\in \{ 1,\ldots ,m\}\), we have \( \nabla \partial _i D^{\alpha '}v \in L^2_{\beta }(B^+_{R}(x_0);{\mathbb {R}}^{(m+1)n})\). If \(\vert \alpha '\vert = l \) and \(\alpha '_{m+1} = 0\) and
then, for every \(B^+_r(y) \in {\mathcal {B}}^+(x_0,R,\tau R)\), either
or
Proof
We use a blow-up argument, analogous in spirit to the argument we used in the proof of Lemma 4.18. First we note that the statement of the lemma is invariant under rescaling and translation by any point in \(\partial {\mathbb {R}}^{m+1}_+\). In particular, suppose the lemma holds for minimisers of \(E^{\beta }\) relative to \(\mathcal {{\tilde{O}}}\subset \partial {\mathbb {R}}^{m+1}_+\) whenever \(\overline{\partial ^0B^+_1(0)}\subset \mathcal {{\tilde{O}}}\). If the hypotheses of the lemma hold for minimisers of \(E^{\beta }\) relative to \({\mathcal {O}}\) and \(B^+_R(x_0)\) satisfies \(R \le 1\) and \(\overline{\partial ^0B^+_R(x_0)}\subset {\mathcal {O}}\), then applying the lemma to \(v_R:= v(R \cdot + x_0)\) yields the conclusion of the lemma on \(B^+_R(x_0)\).
We now prove the lemma when \(R = 1\), \(x_0 = 0\) and v is a minimiser of \(E^{\beta }\) relative to \({\mathcal {O}}\) and \(\overline{\partial ^0B^+_1(0)}\subset {\mathcal {O}}\). Suppose the statement is false. Then there exists \(\delta >0\) such that, for any fixed \(\theta \in (0,\frac{1}{4}]\), we may find a sequence \((v_k)_{k\in {\mathbb {N}}}\) of minimisers of \(E^{\beta }\) relative to \({\mathcal {O}}\) such that the following holds. Each \(v_k\) satisfies \(D^{\alpha '}v_k \in C^{0,\gamma }(\overline{B^+_1(0 )};{\mathbb {R}}^n) \cap {W}^{1,2}_{\beta }(B^+_1( 0);{\mathbb {R}}^n)\) for a \(\gamma \in (0,1)\) and \(\nabla D^{\alpha '}v_k \in L^{\infty }(B^+_1( 0);{\mathbb {R}}^{(m+1)n})\) for every multi-index \(\alpha ' \in {\mathbb {N}}^{m+1}_0\) with \(\alpha '_{m+1} = 0\) and \(\vert \alpha '\vert \le l\). For \(\alpha ' \in {\mathbb {N}}^{m+1}_0\) with \(\alpha '_{m+1} = 0\) and \(\vert \alpha '\vert = l\) and a fixed \(i = 1,\ldots ,m\), each \(v_k\) further satisfies \( \nabla \partial _i D^{\alpha '}v_k \in L^2_{\beta }(B^+_{1}( 0);{\mathbb {R}}^{(m+1)n})\). Moreover, the \(v_k\) satisfy
and, furthermore, there exists a sequence of numbers \( 0< \tau _k \rightarrow 0^+\), half-balls \(B^+_{ r_k}(y_k) \in {\mathcal {B}}^+(0,1,\tau _k)\), and numbers \(0<r_k\le \tau _k \rightarrow 0^+\) such that
and
for \(\alpha '\) with \(\vert \alpha '\vert = l\) and \(\alpha '_{m+1}=0\).
Since each \(v_k\) is a minimiser of \(E^{\beta }\) relative to \({\mathcal {O}}\) and \(v_k \in W^{1,2}_{\beta }(B^+_{1}(0);{\mathbb {R}}^n)\cap C^{0,\gamma }(\overline{B^+_{1}(0)};{\mathbb {R}}^n)\) we deduce from Lemma 4.23 that each \(v_k\) satisfies
The assumptions of the lemma guarantee that we may apply Lemma 4.26 on \( B^+_\frac{1}{6}(0)\) with \(\lambda = 0\). We do so and conclude that there are constants \(C=C(m,N,\beta )\) and \(C_1,C_2\), which depend on \(m,N,\beta \) and are comprised of a polynomial function, with no constant terms, of \(\vert \vert \nabla v_k\vert \vert _{L^{\infty }(B^+_{\frac{1}{3}}(0);{\mathbb {R}}^{(m+1)n})}\) and consequently satisfy \(C_1,C_2 \rightarrow 0\) as \(k \rightarrow \infty \), such that for \(\alpha '\) with \(\vert \alpha '\vert = 1\) and \(\alpha '_{m+1} = 0\)
as \(k \rightarrow \infty \). If \(l \ge 1\), integrating by parts in (4.1), we see that when \(\vert \alpha '\vert = 1\) and \(\alpha '_{m+1}=0\), \(D^{\alpha '}v_k\) satisfies
for every \(\psi \in C_0^{\infty }(B_{\frac{1}{6}}(0);{\mathbb {R}}^n)\) where, by Young’s inequality, \(\vert G(x,q)\vert \le C_3\vert q\vert ^2 + C_4\) for constants \(C_3 = C_3(m,N,\beta )\) and \(C_4\) which depends on \(m,N,\beta \) and is comprised of a polynomial function, with no constant term, of \(\vert \vert \nabla v_k\vert \vert _{L^{\infty }(B^+_{\frac{1}{ 3}}(0);{\mathbb {R}}^{(m+1)n})}\) and hence \(C_4 \rightarrow 0\) as \(k \rightarrow \infty \). Now recall that by assumption \( D^{\alpha '}v_k \in C^{0,\gamma }(\overline{B^+_{ \frac{1}{6}}(0)};{\mathbb {R}}^{(m+1)n})\cap W^{1,2}_{\beta }(B^+_{ \frac{1}{6}}(0);{\mathbb {R}}^{(m+1)n})\) for each \(\alpha ' \in {\mathbb {N}}^{m+1}\) with \(\vert \alpha '\vert = 1\) and \(\alpha '_{m+1}=0\). Hence applying Lemma 4.23 again in conjunction with (4.86) and (4.87), we deduce that there exist positive constants \({\tilde{C}}_3 = {\tilde{C}}_3(m,N,\beta )\) and \({\tilde{C}}_4\) which depends on \(m,N,\beta \) and k with \({\tilde{C}}_4 \rightarrow 0\) as \(k \rightarrow \infty \) such that
as \(k \rightarrow \infty \). Repeating the preceding process for \(D^{\alpha '}v_k\) with \(\vert \alpha '\vert = 2\), then \(\vert \alpha '\vert = 3, \ldots , \vert \alpha '\vert = l\), we see that
as \(k \rightarrow \infty \) for every \(\alpha '\) with \(\vert \alpha '\vert \le l\) and \(\alpha '_{m+1}=0\).
Now fix \(\alpha '\) with \(\vert \alpha '\vert = l\) and \(\alpha '_{m+1}=0\). Discarding as may \(v_k\) as necessary and re-indexing the resulting sequence we may assume that \(2\tau _k \le 6^{-(l +1)}\) so that \(B^+_{r_k}(y_k) \subset B^+_{2r_k}(y_k) \subset B^+_{2\tau _k}(y_k) \in {\mathcal {B}}^+(0,\frac{1}{3} 6^{-l}, 6^{- (l +1)})\) and, in particular, \( B^+_{2\tau _k}(y_k) \subset B^+_{\frac{1}{3} 6^{-l}}(0) \). Define
Note that it is possible to show, combining Lemma 4.26 with \(\lambda = 0\) and (4.88), that \({\tilde{\varepsilon }}_k^2 \rightarrow 0\), but this is not required in what follows.
We see from (4.84) and (4.85) that
and
Define
Then
Hence, using the change of variables \(x\mapsto r_kx+y_k\), we find
Furthermore, after changing variables again, we deduce from (4.90) that
The combination of (4.92) and the Poincaré inequality for \(A_2\) weights implies \((w_k)_{k \in {\mathbb {N}}}\) is bounded in \(W^{1,2}_{\beta }(B^+_1(0);{\mathbb {R}}^n)\). The the Rellich Compactness lemma, Lemma 2.5, therefore yields a subsequence \((w_{k_j})_{j \in {\mathbb {N}}}\) which converges weakly in \( W^{1,2}_{\beta }(B^+_1(0);{\mathbb {R}}^n)\) and strongly in \(L^2_{\beta }(B^+_1(0);{\mathbb {R}}^n)\) to a \(w \in W^{1,2}_{\beta }(B^+_1(0);{\mathbb {R}}^n)\).
Now we show that w is a weak solution of the Neumann-type problem (2.11) in \(B^+_1(0)\). Let \(\phi \in C^{\infty }_0(B_1(0);{\mathbb {R}}^n)\) and define \({\tilde{\phi }} \in C^{\infty }_0(B_{r_k}(y_k);{\mathbb {R}}^n)\) by \({\tilde{\phi }}(z) = \phi \left( \frac{z-y_k}{r_k}\right) \). We observe that \(r_k\nabla {\tilde{\phi }}(z) =\nabla \phi (x)\) where \(x\in B_1(0)\) and \(z\in B_{r_k}(y_k)\) satisfy \(z = r_kx+y_k\). Hence, using the change of variables \(x\mapsto r_kx+y_k\) and (4.91), we find
As \({\tilde{\phi }} \in C^{\infty }_0(B_{r_k}(y_k);{\mathbb {R}}^n)\), \(v_k\) is a minimiser of \(E^{\beta }\) relative to \({\mathcal {O}}\) and, in view of (4.89), we have \(\frac{r_k^2}{{\tilde{\varepsilon }}_k^2}< \frac{1}{\delta }\), it follows that
where C depends on \(m,N,\beta \), \(\vert \vert {\tilde{\phi }}\vert \vert _{L^{\infty }(B^+_{r_k}(y_k);{\mathbb {R}}^n)} =\vert \vert \phi \vert \vert _{L^{\infty }(B^+_{1}(0);{\mathbb {R}}^n)}\) and is comprised of a polynomial, with no constant terms, of \(\vert \vert \nabla D^{{\tilde{\alpha }}'}v_k\vert \vert _{L^{\infty }(B^+_{\frac{1}{3} 6^{-l}}(0);{\mathbb {R}}^n)}\) where \( \vert {\tilde{\alpha }}' \vert \le l =\vert \alpha '\vert \) and \({\tilde{\alpha }}'_{m+1} = 0\) and is therefore independent of k in view of (4.88).
We combine (4.92) and (4.95) to see that, for any \(\phi \in C^{\infty }_0(B_{1}(0 );{\mathbb {R}}^n)\), the weak convergence of \( w_{k_j}\) to w in \(W^{1,2}_{\beta }(B^+_1(0);{\mathbb {R}}^n)\) yields
since \(r_{k_j} \rightarrow 0\). Hence w is a weak solution of (2.11) in \(B^+_1(0)\).
Using the Rellich Compactness Lemma, Lemma 2.5, we take limits in (4.92) and (4.93) to see that
and
respectively. Now, in view of (4.96), the Poincaré inequality for \(A_2\) weights yields
Lastly, since w is a weak solution of (2.11) we may apply Corollary 2.7 to w with \(\theta \le \frac{1}{4}\) (so that \(2\theta \le \frac{1}{2}\)). This gives a positive constant C (independent of \(\theta \)) and a \(\gamma \in (0,1)\) such that
Combining (4.98) and (4.99) we see that
This holds for all fixed \(\theta \in (0,\frac{1}{4}]\) and we choose \(\theta < 2^{-1}\left( \frac{\delta }{C}\right) ^{\frac{1}{2\gamma }}\) so that (4.100) contradicts (4.97). Hence the lemma is proved. \(\square \)
4.16 Control of the mean squared oscillation of the derivatives in the interior
We need a counterpart to Lemma 4.27 which holds on a class of balls with closure contained in the interior of \({\mathbb {R}}^{m+1}_+\).
Lemma 4.28
Fix \(l \in {\mathbb {N}}_0\). For every \(\delta > 0\) there exist numbers \(\varepsilon >0\), \(\tau \in (0,1)\) and \(\theta \in (0,\frac{1}{4}]\) such that the following holds. Suppose \(v \in {\dot{W}}^{1,2}_{\beta }({\mathbb {R}}^{m+1}_+;N)\) is a minimiser of \(E^{\beta }\) relative to \({\mathcal {O}}\) with \(D^{\alpha '}v \in C^{0,\gamma }(\overline{B^+_R(x_0)};{\mathbb {R}}^n) \cap {W}^{1,2}_{\beta }(B^+_R(x_0);{\mathbb {R}}^n)\) for a \(\gamma \in (0,1)\) and \(\nabla D^{\alpha '}v \in L^{\infty }(B^+_R(x_0);{\mathbb {R}}^{(m+1)n})\) for every multi-index \(\alpha ' \in {\mathbb {N}}^{m+1}_0\) with \(\alpha '_{m+1} = 0\) and \(\vert \alpha '\vert \le l\), where \(B^+_{R}(x_0 ) \subset {\mathbb {R}}^{m+1}_+\) satisfies \(R \le 1\) and \(\overline{\partial ^0 B^+_R(x_0)} \subset {\mathcal {O}}\). Suppose further that for \(\alpha ' \in {\mathbb {N}}^{m+1}_0\) with \(\alpha '_{m+1} = 0\) and \(\vert \alpha '\vert = l\) and some \(i\in \{ 1,\ldots ,m\}\), we have \( \nabla \partial _i D^{\alpha '}v \in L^2_{\beta }(B^+_{R}(x_0);{\mathbb {R}}^{(m+1)n})\). If \(\vert \alpha '\vert = l \) and \(\alpha '_{m+1} = 0\) and
then, for every \(B_r(y) \in {\mathcal {B}}_4(x_0,R,\tau R)\), either
or
Proof
The method of proof is similar to the proof of Lemma 4.27. We observe that the lemma is invariant under scaling and translation with respect to \(x_0\) in \(\partial {\mathbb {R}}^{m+1}_+\) in the same way as Lemma 4.27. Hence we assume \(R = 1\), \(x_0 = 0\), v is a minimiser of \(E^{\beta }\) relative to \({\mathcal {O}}\) and \(\overline{\partial ^0B^+_1(0)}\subset {\mathcal {O}}\).
Suppose the statement is false. Then there exists \(\delta >0\) such that, for any fixed \(\theta \in (0,\frac{1}{4}]\), we may find a sequence \((v_k)_{k\in {\mathbb {N}}}\) of minimisers of \(E^{\beta }\) relative to \({\mathcal {O}}\) such that the following holds. Each \(v_k\) satisfies \(D^{\alpha '}v_k \in C^{0,\gamma }(\overline{B^+_1(0 )};{\mathbb {R}}^n) \cap {W}^{1,2}_{\beta }(B^+_1( 0);{\mathbb {R}}^n)\) for a \(\gamma \in (0,1)\) and \(\nabla D^{\alpha '}v_k \in L^{\infty }(B^+_1( 0);{\mathbb {R}}^{(m+1)n})\) for every multi-index \(\alpha ' \in {\mathbb {N}}^{m+1}_0\) with \(\alpha '_{m+1} = 0\) and \(\vert \alpha '\vert \le l\). For \(\alpha ' \in {\mathbb {N}}^{m+1}_0\) with \(\alpha '_{m+1} = 0\) and \(\vert \alpha '\vert = l\) and a fixed \(i = 1,\ldots ,m\), each \(v_k\) further satisfies \( \nabla \partial _i D^{\alpha '}v_k \in L^2_{\beta }(B^+_{1}( 0);{\mathbb {R}}^{(m+1)n})\). The \(v_k\) also satisfy
There furthermore exists a sequence of numbers \( 0< \tau _k \rightarrow 0\), balls \(B_{ r_k}(y_k) \in {\mathcal {B}}_4(0,1,\tau _k)\), and numbers \(0<r_k\le \tau _k \rightarrow 0\) such that
and
for \(\alpha '\) with \(\vert \alpha '\vert = l\) and \(\alpha '_{m+1}=0\).
Since the assumptions of the lemma are the same as the assumptions of Lemma 4.27 we still have (4.88), namely, for every \(\alpha '\) with \(\vert \alpha '\vert \le l\) and \(\alpha '_{m+1}=0\)
Define
Fix \(\alpha '\) with \(\vert \alpha '\vert = l\) and \(\alpha '_{m+1} = 0\). Discarding as many k as necessary, and re-indexing the resulting sequence to \(k \in {\mathbb {N}}\), we may assume \(2\tau _k \le 6^{-(l +1)}\) so that \(B_{r_k}(y_k) \in {\mathcal {B}}_4(0,\frac{1}{3} 6^{-l}, 6^{- (l +1)})\) and, in particular, \( B_{r_k}(y_k) \subset B^+_{\frac{1}{3} 6^{-l}}(0) \). Note that similarly to in the proof of Lemma 4.27, using Lemma 4.26 with \(\lambda = 0\) and (4.105) we can show \({\tilde{\varepsilon }}_k^2 \rightarrow 0\), but this is not used in what follows. Now consider the normalised sequence
We have
Hence, using the change of variables \(x \mapsto {r_kx+y_k}\), we find
and, also using (4.104),
As a result of (4.107) and the Poincaré inequality, we observe \((w_k)_{k \in {\mathbb {N}}}\) is bounded in \(W^{1,2}(B_{1}( 0);{\mathbb {R}}^n)\). The Rellich Compactness lemma [39, Section 1.3, Lemma 1], thus yields a subsequence \((w_{k_j})_{j \in {\mathbb {N}}}\) which converges weakly in \( W^{1,2}(B_{1}( 0);{\mathbb {R}}^n)\) and strongly in \(L^2(B_{1}( 0);{\mathbb {R}}^n)\) to some \(w\in W^{1,2}(B_{1}(0);{\mathbb {R}}^n)\).
Define \(f_{k}(x) = \left( 1 + (y_{k})_{m+1}^{-1} r_{k} x_{m+1}\right) ^{\beta }\) for each \(k \in {\mathbb {N}}\). Observe that \(a_{k_j} = (y_{k_j})_{m+1}^{-1}r_{k_j} \in [0,4^{-1}]\) for every j, since each \(B_{ r_k}(y_k) \in {\mathcal {B}}_4(0,1,\tau _k)\). Thus there is a subsequence, which we also index with \({k_j}\), which converges to \(a \in [0,4^{-1}]\). Furthermore, \((f_{k_j})_{j \in {\mathbb {N}}}\) is uniformly bounded and equicontinuous so, by the Arzelà–Ascoli theorem, there is a uniformly convergent subsequence which we again index by \(k_j\). Since \(f_{k_j}(x) \rightarrow f(x) = \left( 1 + ax_{m+1}\right) ^{\beta }\) pointwise, we must also have \(f_{k_j} \rightarrow f\) uniformly.
Now, for \(\phi \in C^{\infty }_0(B_1(0);{\mathbb {R}}^n)\), similar calculations to those in the proof of Lemma 4.27 yield
as \(k \rightarrow \infty \). Furthermore, as \( w_{k_j}\) converges weakly to w in \(W^{1,2}(B_1(0);{\mathbb {R}}^n)\) and \(f_{k_j} \rightarrow f\) uniformly, we conclude that
Hence w is a weak solution of \(\text {div}(\left( 1 + ax_{m+1}\right) ^{\beta } \nabla w) = 0 \) in \(B_{1}(0)\). By linear elliptic regularity theory, w is smooth in \(B_1(0)\). We also conclude by taking limits in (4.107) and (4.108) that
respectively using the Rellich Compactness Lemma. Since \(\vert \nabla w\vert ^2\) satisfies a mean value inequality, namely \(\sup _{B_{\theta }( 0)}\vert \nabla w\vert ^2 \le C(m,\beta )\int _{B_{1}( 0)} \vert \nabla w \vert ^2 \mathrm {d}x\) as shown in Theorem 2.1 in section III of [14], we apply the Poincaré inequality and (4.111) to see that
This holds for all fixed \(\theta \in (0,\frac{1}{2}]\) and we choose \(\theta < \left( \frac{\delta }{C }\right) ^{\frac{1}{2}}\) so that (4.112) contradicts the last statement of (4.111). \(\square \)
4.17 Higher order \(\varepsilon \)-regularity
With the preceding theory in hand, we are now in a position to prove our main \(\varepsilon \)-regularity theorem.
Proof of Theorem 4.1
Observe that the hypothesis of the theorem are invariant with respect to the rescaling \(x \mapsto Rx+x_0\). Thus we will assume \(R = 1, x_0 = 0\) and \(\overline{\partial ^0B^+_1(0)}\subset \mathcal {{O}}\).
We use proof by strong induction. We choose \(\varepsilon \) to be the number from Theorem 4.21. Then the combination of Theorem 4.21, Lemmata 4.23 and 4.25 yield a \({\hat{\theta }} = {\hat{\theta }}(m,N,\beta ) \le \frac{1}{2}\) and a \({\hat{\gamma }} \in (0,1)\) such that \(v \in C^{0,{\hat{\gamma }}}(\overline{B^+_{{\hat{\theta }}}(0)};{\mathbb {R}}^n) \cap W^{1,2}_{\beta }( B^+_{{\hat{\theta }}}(0);{\mathbb {R}}^n)\), \(\nabla v \in L^{\infty }(B^+_{{\hat{\theta }}}(0);{\mathbb {R}}^{(m+1)n})\) and \(\partial _i v \in W^{1,2}_{\beta }( B^+_{{\hat{\theta }}}(0);{\mathbb {R}}^n)\) for \(i = 1,\ldots ,m\). Now fix \(l \in {\mathbb {N}}_0\). The induction hypothesis is that there exists \({\tilde{\theta }} = {\tilde{\theta }}(m,N,\beta , l) \le \frac{1}{2}\) and a \({\tilde{\gamma }} = {\tilde{\gamma }}(m,N,\beta ,l) \in (0,1)\) such that the following holds. For \(\alpha '\in {\mathbb {N}}^{m+1}_0\) with \(\alpha '_{m+1} = 0\) and \(\vert \alpha '\vert \le l\), we have \( D^{\alpha '}v \in C^{0,{\tilde{\gamma }}}(\overline{B^+_{{\tilde{\theta }}}(0)};{\mathbb {R}}^n)\cap W^{1,2}_{\beta }( B^+_{{\tilde{\theta }}}(0);{\mathbb {R}}^n)\) and \(\nabla D^{\alpha '}v \in L^{\infty }(B^+_{{\tilde{\theta }}}(0);{\mathbb {R}}^{(m+1)n})\). Furthermore, when \(\vert \alpha '\vert = l\) and \(\alpha '_{m+1} = 0\) we suppose \(\nabla \partial _i D^{\alpha '}v \in L^2_{\beta }( B^+_{{\tilde{\theta }}}(0);{\mathbb {R}}^{n(m+1)})\) for \(i = 1,\ldots ,m\). We have already observed that this is true when \(l = 0\). The inductive step will be to show the preceding statement holds, possibly for a different \({\tilde{\theta }}\) and \({\tilde{\gamma }}\), for \(D^{\alpha '}v\) with \(\vert \alpha ' \vert \le l+1\) and \(\alpha '_{m+1} = 0\). We fix \(\alpha '\) with \(\vert \alpha ' \vert = l \ge 0 \) and \(\alpha '_{m+1} = 0\) henceforth.
Applying Lemma 4.26, we see that
for any \(B_{\rho }(y) \subset B_{{\tilde{\theta }}}(0)\) with \(y_{m+1} \ge 0\) and \(i = 1,\ldots ,m\), where \(C_0=C_0(m,N,\beta )\) and \(C_1,C_2\) depend on \(m,N,\beta \) and are polynomial functions, with no constant terms, of \(\vert \vert \nabla D^{{\tilde{\alpha }}'}v\vert \vert _{L^{\infty }(B_\rho (y)\cap {\mathbb {R}}^{m+1}_+;{\mathbb {R}}^{(m+1)n})}\) where \( {\tilde{\alpha }}'\in {\mathbb {N}}^{m+1}_0\) with \(\vert {\tilde{\alpha }}'\vert \le l\) and \({\tilde{\alpha }}_{m+1}' = 0\). We apply Lemmata 4.27 and 4.28, with \(\delta =\frac{1}{2} \frac{1}{2^{m+2}C_0}\), to respectively obtain numbers \(\varepsilon _1>0\), \(\tau _1 \in (0,1)\) and \(\theta _1 \in (0,\frac{1}{4}]\) and \(\varepsilon _2>0\), \(\tau _2 \in (0,1)\) and \(\theta _2 \in (0,\frac{1}{4}]\), depending only on \(\delta \) and hence only on \(m,N,\beta \), such that if \({\tilde{R}} \le {\tilde{\theta }}\) and
then either (4.82) or (4.83) holds for every \(B^+_{r_1}(y) \in {\mathcal {B}}^+(0,{\tilde{R}},\tau _1{\tilde{R}}) \) and either (4.101) or (4.102) holds for every \(B_{r_2}(y) \in {\mathcal {B}}_4(0,{\tilde{R}},\tau _2{\tilde{R}}) \). It follows from the proof of Theorem 4.21, bearing in mind \(R=1\) and \(x_0 = 0\), that for every \(B^+_r(y)\in {\mathcal {B}}^+(0,1,\frac{1}{2})\) we have
for some \({\tilde{\gamma }} \in (0,1)\) and some constant \(C = C(m,N,\beta )\). In particular, this holds for \(y = 0\) and \(r \le \frac{1}{2}\). Hence if \({\tilde{R}} = {\tilde{R}}(m,N,\beta ,l) =( \min \{\frac{\varepsilon _1^2}{C },\frac{\varepsilon _2^2}{C }, \left( \frac{{\tilde{\theta }}}{2}\right) ^{{\tilde{\gamma }}}\})^{\frac{1}{{\tilde{\gamma }}}}\) then (4.114) holds on \(B^+_{{\tilde{R}}}(0)\). We have assumed \({\tilde{R}}\le \frac{{\tilde{\theta }}}{2} \) so that we may later apply (4.113) with impunity on any ball or half-ball in \(B^+_{{\tilde{R}}}(0)\).
First we show that (4.7) essentially holds for \(\nabla \partial _i D^{\alpha '}v\) on every \(B^+_{r_1}(y) \in {\mathcal {B}}^+(0,{\tilde{R}},\tau _1{\tilde{R}})\). We know that (4.82) or (4.83) holds on \(B^+_{r_1}(y)\). We apply (4.113) with \(\lambda = \overline{\partial _iD^{ {\alpha }'}v}_{B^+_{\theta _1r_1}(y),\beta }\), noting that \(\vert \lambda \vert \le \vert \vert \nabla D^{ {\alpha }'}v\vert \vert _{L^{\infty }(B^+_{\theta _1r_1}(y);{\mathbb {R}}^{(m+1)n})}\), to see that
Hence, regardless of which of (4.82) or (4.83) holds (bearing in mind our choice of \(\delta \) above), we have
where \(\sigma _1 = \frac{\theta _1 }{2}\) and C depends on \(\vert \vert \nabla D^{{\tilde{\alpha }}'}v \vert \vert _{L^{\infty }(B^+_{{\tilde{\theta }}}(0);{\mathbb {R}}^{(m+1)n})}\) where \(\vert {\tilde{\alpha }}'\vert \le l\) with \({\tilde{\alpha }}'_{m+1}=0\) and, moreover, may depend on \({\tilde{R}},\theta _1,m,N\) and \(\beta \) and hence only on \(m,N,\beta ,l\) as \({\tilde{R}} = {\tilde{R}}(m,N,\beta ,l)\) and \(\theta _1 = \theta _1(m,N,\beta )\). This holds for any \(B^+_{r_1}(y) \in {\mathcal {B}}^+(0,{\tilde{R}},\tau _1{\tilde{R}})\). We may apply (4.116) with \(r_1\) replaced by \(\sigma _1^kr_1\) for every \(k \in {\mathbb {N}}\) and iterate to see that
Setting \(\gamma _1 = -\frac{\ln 2}{\ln \sigma _1} \in (0,1)\) we conclude that
for any \(r \le r_1\), where C depends on the same factors as the constant in (4.116). This holds for any \(B^+_{r_1}(y) \in {\mathcal {B}}^+(0,{\tilde{R}},\tau _1{\tilde{R}})\).
We want a similar estimate for \({r_2}^{1-m}\int _{B_{r_2}(y)} \vert \nabla \partial _iD^{ {\alpha }'}v\vert ^2\mathrm {d}x\) on balls \(B_{r_2}(y) \in {\mathcal {B}}_4(0,{\tilde{R}},\tau _2{\tilde{R}})\). We calculate the constants in (4.6) from Sect. 4.3 explicitly. Then our choice of \(\delta \) and a similar argument which lead to (4.118) yields the existence of a \(\gamma _2 = \gamma _2(m,N,\beta ) \in (0,1)\) such that for any \(B_{r_2}(y) \in {\mathcal {B}}_4(0,{\tilde{R}},\tau _2{\tilde{R}})\) and any \(r \le r_2\) we have
where \(\sigma _2 = \frac{\theta _2}{2}\) and C depends on the same factors as in (4.118). We now use (4.118) and (4.119) to show the hypothesis (4.7) from Lemma 4.8 is satisfied.
Define \(\tau = \min \{\frac{ \tau _1}{2},\tau _2\}< \frac{1}{2}\), \(\gamma = \min \{\gamma _1,\gamma _2\}\). We apply (4.118) with \(r_1 = \tau {\tilde{R}}\). It follows that for every \(B^+_r(y) \in {\mathcal {B}}^+(0,{\tilde{R}},\tau {\tilde{R}})\) we have \(B^+_{\tau {\tilde{R}}}(y) \in {\mathcal {B}}^+(0,{\tilde{R}},\tau _1 {\tilde{R}})\) and hence
Furthermore, applying (4.113) with \(\lambda = 0\) and \(\rho = 2\tau {\tilde{R}}\) implies that
where C depends on \(m,N,\beta ,l\) and \(\vert \vert \nabla D^{{\tilde{\alpha }}'}v \vert \vert _{L^{\infty }(B^+_{{\tilde{\theta }}}(0);{\mathbb {R}}^{(m+1)n})}\) where \(\vert {\tilde{\alpha }}'\vert \le l\) and \({\tilde{\alpha }}'_{m+1} = 0\). We combine (4.120) and (4.121) to see that for every \(B^+_r(y) \in {\mathcal {B}}^+(0,{\tilde{R}},\tau {\tilde{R}})\) we have
where \(C_3\) depends on \(m,N,\beta ,l\) and \(\vert \vert \nabla D^{{\tilde{\alpha }}'}v \vert \vert _{L^{\infty }(B^+_{{\tilde{\theta }}}(0);{\mathbb {R}}^{(m+1)n})}\) where \(\vert {\tilde{\alpha }}'\vert \le l\) and \({\tilde{\alpha }}'_{m+1} = 0\).
Now let \(B_r(y) \in {\mathcal {B}}_4(0,{\tilde{R}},\frac{2\tau }{3}{\tilde{R}})\). Then \(B_r(y) \subset B_{\frac{y_{m+1}}{4}}(y)\subset B^+_{\frac{3y_{m+1}}{2}}(y^+)\subset B^+_{\tau {\tilde{R}}}(y^+) \in {\mathcal {B}}^+(0,{\tilde{R}},\tau {\tilde{R}})\), where \(y^+ = y - (0,y_{m+1})\). Recalling again (4.6) from Sect. 4.3, we note that
Since \(y_{m+1} \le \tau {\tilde{R}}\le 1\), applying (4.119) on \(B_{\frac{y_{m+1}}{4}}(y) \in {\mathcal {B}}_4(0,{\tilde{R}},\tau _2{\tilde{R}})\), using (4.123), and then applying (4.122) gives
where \(C_4\) depends on \(m,N,\beta ,l\) and \(\vert \vert \nabla D^{{\tilde{\alpha }}'}v \vert \vert _{L^{\infty }(B^+_{{\tilde{\theta }}}(0);{\mathbb {R}}^{(m+1)n})}\) where \(\vert {\tilde{\alpha }}'\vert \le l\) and \({\tilde{\alpha }}'_{m+1} = 0\). Together, (4.122) and (4.124) imply (4.7) from Lemma 4.8 holds for \(B^+_r(y) \in {\mathcal {B}}^+(0,{\tilde{R}},\frac{2\tau }{3}{\tilde{R}})\) and \(B_r(y) \in {\mathcal {B}}_4(0,{\tilde{R}},\frac{2\tau }{3}{\tilde{R}})\). Applying this lemma shows that \(\partial _i D^{\alpha '}v \in C^{0,{\hat{\gamma }}}(\overline{B^+_{{\hat{\theta }}}(0)};{\mathbb {R}}^n)\) for some \({\hat{\theta }},{\hat{\gamma }} \in (0,1)\) depending on \(m,N,\beta ,l\). Hence, recalling that the inductive hypothesis implies that \(\partial _iD^{\alpha '}v \in W^{1,2}_{\beta }(B^+_{{\tilde{\theta }}}(0) ;{\mathbb {R}}^n)\), we may apply Lemmata 4.23 and 4.25 to respectively imply \(\nabla \partial _iD^{\alpha '}v \in L^{\infty }(B^+_{\frac{{\hat{\theta }}}{3}} (0);{\mathbb {R}}^{(m+1)n})\) and \(\nabla \partial _j\partial _i D^{\alpha '} v \in L^2_{\beta }( B^+_{\frac{{\hat{\theta }}}{6}}(0);{\mathbb {R}}^{n(m+1)})\) for \(i,j = 1,\ldots ,m\). This completes the inductive step and therefore the proof. \(\square \)
Remark 4.29
A consequence of the proof is that the number \(\varepsilon \) in Theorem 4.1 can be taken to be the number from Theorem 4.21.
Theorem 4.1 yields an improvement to Theorem 4.21 for minimisers of \(E^{\beta }\) relative to \({\mathcal {O}}\). We are now in a position to prove our partial regularity theorem.
Proof of Theorem 4.3
Aspects of the proof closely follow the proof of Theorem 3.2 in [29]. First, it follows from the theory of Schoen and Uhlenbeck [37] that there exists a set \({\varSigma }_{\text {int}}\subset {\mathbb {R}}^{m+1}_+\), with Hausdorff dimension at most \(m-2\), such that v is smooth in a neighbourhood of any point in \({\varSigma }_{\text {int}}\).
Define
where \(\varepsilon \) is the number given by the Theorem 4.21 and \( {\varTheta }^{\beta }_v(y) \) is the density function defined in Remark 4.6. The upper semi-continuity of \({\varTheta }^{\beta }_v\) was established in Remark 4.6 which, when combined with the definition of \({\varSigma }_{\text {bdry}}\), shows that \({\varSigma }_{\text {bdry}}\) is relatively closed in \({\mathcal {O}}\).
We write \({\varSigma }_{\text {bdry}}\) as a countable union of compact sets of the form \(K\cap {\varSigma }_{\text {bdry}}\), where \(K\subset {\mathcal {O}}\) is compact, and let \({\varSigma }'\subset {\varSigma }_{\text {bdry}}\) be such a set. Fix \(\delta > 0\) and cover \({\varSigma }'\) by a collection of balls \(B^m_{r_i}(x_i)\subset {\mathcal {O}}\) with \(\overline{B^m_{r_i}(x_i)}\subset {\mathcal {O}}\) with \(x_i \in {\varSigma }'\) and \(0<r_i \le \delta \). The compactness of \({\varSigma }'\), combined with Vitali’s covering theorem yields a finite subcollection of balls, \(B^m_{r_1}(x_1), \ldots , B^m_{r_I}(x_I)\) for some \(I \in {\mathbb {N}}\), of any such cover of \({\varSigma }'\), which satisfies \( B_{r_i}(x_i)\cap B_{r_j}(x_j) = \emptyset \quad \text {for} \ i \ne j, 1\le i,j\le I\quad \text {and}\quad {\varSigma }'\subset \bigcup _{i=1}^I B_{5r_i}(x_i).\) Using the boundary energy monotonicity formula, Lemma 4.5, we see that
We send \(\delta \rightarrow 0^+\) and use Lebesgue’s Dominated Convergence Theorem to see that \({\mathcal {H}}^{m-1+\beta }({\varSigma }') = 0\) and hence \({\mathcal {H}}^{m-1+\beta }({\varSigma }_{\text {bdry}}) = 0\).
Let \(x_0 \in ({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}){\backslash } {\varSigma }\). If \(x_0 \in {\mathbb {R}}^{m+1}_+\) then \(x_0 \in {\mathbb {R}}^{m+1}_+{\backslash } {\varSigma }_{\text {int}}\) and v is smooth in an open ball centred at \(x_0\) and contained in \({\mathbb {R}}^{m+1}_+{\backslash } {\varSigma }_{\text {int}}\subset ({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}){\backslash }{\varSigma }\). If \(x_0 \in {\mathcal {O}}\) then \(x_0 \in {\mathcal {O}}{\backslash } {\varSigma }_{\text {bdry}}\) and \({\varTheta }^{\beta }_v(x_0) <\varepsilon \) which, combined with the fact that \({\mathcal {O}}{\backslash } {\varSigma }_{\text {bdry}}\) is open in \({\mathcal {O}}\), implies there exists an \(R>0\) such that \(R^{1-m-\beta }\int _{B^+_R(x_0)}x_{m+1}^{\beta }\vert \nabla v \vert ^2\mathrm {d}x \le \varepsilon \), \(R \le 1\) and \(\overline{\partial ^0B^+_R(x_0)} \subset {\mathcal {O}}{\backslash } {\varSigma }_{\text {bdry}}\). Consequently, Theorem 4.21 implies that there are \(\theta ,\gamma \in (0,1)\) such that \(v \in C^{0,\gamma }(\overline{B^+_{\theta R}(x_0)};N)\). Furthermore, we deduce from (4.61) in the proof of Theorem 4.21 that
on every \(B^+_{r}(z) \in {\mathcal {B}}^+(x_0,R,\frac{R}{2})\) which shows that \({\varTheta }_v^{\beta }(z) = 0\) for every \(z \in \partial ^0B^+_{\frac{R}{2}}(x_0)\). Now setting \(\sigma = \min \{\theta ,\frac{1}{2}\}\) we see that \({\varTheta }_v^{\beta }(z) = 0\) for \(z \in \partial ^0B^+_{\sigma R}(x_0)\) which implies \(\partial ^0B^+_{\sigma R}(x_0) \subset {\mathcal {O}}{\backslash }{\varSigma }_{\text {bdry}}\). Furthermore, v is a Hölder continuous weakly harmonic map in any \(B_r(y)\) with \(\overline{B_r(y)} \subset B^+_{\sigma R}(x_0)\). We apply Lemma 4.13 to see that v is smooth in \(B^+_{\sigma R}(x_0)\) and conclude that \(B^+_{\sigma R}(x_0)\subset {\mathbb {R}}^{m+1}_+{\backslash }{\varSigma }_{\text {int}}\). Consequently, we have \(B^+_{\sigma R}(x_0) \cup \partial ^0B^+_{\sigma R}(x_0) \subset ({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}){\backslash } {\varSigma }\). Note that \(B^+_{\sigma R}(x_0) \cup \partial ^0B^+_{\sigma R}(x_0)\) is an open ball centred at \(x_0\) in the (Euclidean) topology of \({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}\). Hence \({\varSigma }\) is relatively closed in \({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}\). As the Hausdorff dimension of \({\varSigma }_{int}\) is at most \(m-2<m-1+\beta \) and \({\mathcal {H}}^{m-1+\beta }({\varSigma }_{\text {bdry}}) = 0\), we deduce that \({\mathcal {H}}^{m +\beta - 1}({\varSigma }) = 0\). We also conclude \(v \in C^{0,\gamma }_{loc}(({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}){\backslash } {\varSigma };N)\).
Consider \(x_0 \in ({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}){\backslash } {\varSigma }\) with \(x_0 \in {\mathcal {O}}\). Then, as above, we observe there is an \(R>0\) such that \(R^{1-m-\beta }\int _{B^+_R(x_0)}x_{m+1}^{\beta }\vert \nabla v \vert ^2\mathrm {d}x \le \varepsilon \), \(R \le 1\) and \(\overline{\partial ^0B^+_R(x_0)} \subset {\mathcal {O}}{\backslash } {\varSigma }_{\text {bdry}}\). In view of Remark 4.29, Theorem 4.1 implies that for every \(l \in {\mathbb {N}}_0\) there exist \(\theta ,\gamma \in (0,1)\) such that for every \(\alpha ' \in {\mathbb {N}}^{m+1}_0\) with \(\vert \alpha '\vert \le l\) and \(\alpha '_{m+1} = 0\) we have \( D^{ {\alpha }'}v \in C^{0,\gamma }(\overline{B^+_{\theta R}(x_0)};{\mathbb {R}}^n)\). However, we also know that \(({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}){\backslash }{\varSigma }\) is open in \({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}\). Hence there exists \({\tilde{R}}>0\) such that \(B^+_{{\tilde{R}}}(x_0)\cup \partial ^0B^+_{{\tilde{R}}}(x_0)\subset ({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}){\backslash }{\varSigma }\). Setting \(r = \min \{\theta R,{\tilde{R}}\}\) we conclude \( D^{ {\alpha }'}v \in C^{0,\gamma }(\overline{B^+_{r}(x_0)};{\mathbb {R}}^n)\) and \(B^+_{r}(x_0)\cup \partial ^0B^+_{r}(x_0)\subset ({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}){\backslash }{\varSigma }\). We iteratively apply Lemmata 4.23 and 4.25 to see that for every \(\alpha ' \in {\mathbb {N}}^{m+1}_0\) with \(\vert \alpha '\vert \le l\) and \(\alpha '_{m+1} = 0\) we have \(\nabla D^{\alpha '}v \in L^{\infty }(B^+_{{\tilde{r}}}(x_0);{\mathbb {R}}^{(m+1)n})\) and \(D^{\alpha '}v \in W^{1,2}_{\beta }(B^+_{{\tilde{r}}}(x_0);{\mathbb {R}}^n)\) for some \({\tilde{r}} \le r\). It follows that \(B^+_{{\tilde{r}}}(x_0)\cup \partial ^0B^+_{{\tilde{r}}}(x_0)\subset ({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}){\backslash }{\varSigma }\). Hence \(v \in C^{0,1}_{loc}(({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}){\backslash } {\varSigma };N)\) and for every multi-index \(\alpha ' \in {\mathbb {N}}^{m+1}\) with \(\alpha '_{m+1} = 0\) we have \(D^{\alpha '}v \in C^{0,1}_{loc}(({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}){\backslash } {\varSigma };{\mathbb {R}}^n)\) and \(\nabla D^{\alpha '}v \in L^{\infty }_{loc}(({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}){\backslash } {\varSigma };{\mathbb {R}}^{(m+1)n})\).
Lastly, for \(x_0\) as in the preceding paragraph, fix \(\alpha ' \in {\mathbb {N}}^{m+1}_0\) with \(\alpha '_{m+1} = 0\) and \({\tilde{r}}\le 1\) such that \(B^+_{{\tilde{r}}}(x_0)\cup \partial ^0B^+_{{\tilde{r}}}(x_0)\subset ({\mathbb {R}}^{m+1}_+\cup {\mathcal {O}}){\backslash }{\varSigma }\). Making \({\tilde{r}}\) smaller if necessary, we may assume that \(\nabla D^{{\tilde{\alpha }}'}v \in L^{\infty }(B^+_{{\tilde{r}}}(x_0);{\mathbb {R}}^{(m+1)n})\) and \(D^{\tilde{\alpha '}}v \in C^{0,1}(\overline{B^+_{{\tilde{r}}}(x_0)};{\mathbb {R}}^n)\) for every \({\tilde{\alpha }}' \in {\mathbb {N}}^{m+1}_0\) with \({\tilde{\alpha }}'_{m+1} = 0\) and \(\vert \tilde{\alpha '}\vert \le \vert \alpha '\vert +2\). We also observe that since \(v \in C^{0,1}(\overline{ B^+_{{\tilde{r}}}(x_0)};N)\) is a Hölder continuous harmonic map, it is smooth in \(B^+_{{\tilde{r}}}(x_0)\) by Lemma 4.13 and so we have \(x_{m+1}^{-\beta }\partial _{m+1} (x_{m+1}^{\beta }\partial _{m+1}D^{\alpha '}v) = -({\varDelta }'D^{\alpha '}v + D^{\alpha '}(A(v)(\nabla v,\nabla v)))\) classically in \(B^+_{{\tilde{r}}}(x_0)\), where \({\varDelta }'\) is the Laplacian with respect to \(x_i\), \(i = 1,\ldots ,m\). Hence, \(x_{m+1}^{-\beta }\partial _{m+1} (x_{m+1}^{\beta }\partial _{m+1}D^{\alpha '}v)\) is bounded in \(B^+_{{\tilde{r}}}(x_0)\). Hence, for every \(B^+_{\rho }(y) \in {\mathcal {B}}^+(x_0,\frac{{\tilde{r}}}{3},\frac{{\tilde{r}}}{6})\) we calculate
Moreover, for every \(B_{\rho }(y) \in {\mathcal {B}}(x_0,\frac{{\tilde{r}}}{3},\frac{{\tilde{r}}}{6})\) we calculate
An application of Lemma 4.8 concludes the proof. \(\square \)
References
Bethuel, F.: A new obstruction to the extension problem for Sobolev maps between manifolds. J. Fixed Point Theory Appl. 15(1), 155–183 (2014). https://doi.org/10.1007/s11784-014-0185-0
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)
Caffarelli, L.A., Roquejoffre, J.M., Sire, Y.: Variational problems for free boundaries for the fractional Laplacian. J. Eur. Math. Soc. (JEMS) 12(5), 1151–1179 (2010). https://doi.org/10.4171/JEMS/226
Caffarelli, L.A., Salsa, S., Silvestre, L.: Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Invent. Math. 171(2), 425–461 (2008). https://doi.org/10.1007/s00222-007-0086-6
Cohn, D.L.: Measure Theory, 2nd edn. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser/Springer, New York (2013)
Da Lio, F.: Fractional harmonic maps into manifolds in odd dimension \(n>1\). Calc. Var. Partial Differ. Equ. 48(3–4), 421–445 (2013)
Da Lio, F., Rivière, T.: Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps. Adv. Math. 227(3), 1300–1348 (2011)
Da Lio, F., Rivière, T.: Three-term commutator estimates and the regularity of \(\frac{1}{2}\)-harmonic maps into spheres. Anal. PDE 4(1), 149–190 (2011)
Da Lio, F., Schikorra, A.: \(\frac{n}{p}\)-harmonic maps: regularity for the sphere case. Adv. Calc. Var. 7(1), 1–26 (2014)
Duzaar, F., Steffen, K.: An optimal estimate for the singular set of a harmonic map in the free boundary. J. Reine Angew. Math. 401, 157–187 (1989)
Duzaar, F., Steffen, K.: A partial regularity theorem for harmonic maps at a free boundary. Asymptot. Anal. 2(4), 299–343 (1989)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)
Fabes, E.B., Kenig, C.E., Serapioni, R.P.: The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differ. Equ. 7(1), 77–116 (1982)
Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies, vol. 105. Princeton University Press, Princeton (1983)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1983)
Große-Brauckmann, K.: Interior and boundary monotonicity formulas for stationary harmonic maps. Manuscr. Math. 77(1), 89–95 (1992)
Gulliver, R., Jost, J.: Harmonic maps which solve a free-boundary problem. J. Reine Angew. Math. 381, 61–89 (1987)
Hardt, R., Lin, F.H.: Partially constrained boundary conditions with energy minimizing mappings. Commun. Pure Appl. Math. 42(3), 309–334 (1989)
Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (1993). Oxford Science Publications
Hélein, F.: Regularité des applications faiblement harmoniques entre une surface et une sphere. C. R. Acad. Sci. Paris Sér. I Math. 311(9), 519–524 (1990)
Hélein, F.: Sur la régularité des applications faiblement harmoniques entre une surface et une variété riemannienne. In: Séminaire sur les Équations aux Dérivées Partielles, 1990–1991, pp. Exp. No. X, 6. École Polytech., Palaiseau (1991)
Jost, J.: Riemannian Geometry and Geometric Analysis. Springer, Berlin (2008)
Luckhaus, S.: Convergence of minimizers for the \(p\)-Dirichlet integral. Math. Z. 213(3), 449–456 (1993)
Maz’ya, V.: Sobolev spaces with applications to elliptic partial differential equations. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 342, augmented edn. Springer, Heidelberg (2011)
Millot, V., Sire, Y.: On a fractional Ginzburg–Landau equation and 1/2-harmonic maps into spheres. Arch. Ration. Mech. Anal. 215(1), 125–210 (2014)
Millot, V., Sire, Y., Yu, H.: Minimizing fractional harmonic maps on the real line in the supercritical regime. ArXiv e-prints (2017)
Morrey Jr., C.B.: The problem of Plateau on a Riemannian manifold. Ann. Math. 2(49), 807–851 (1948)
Morrey Jr., C.B.: Multiple Integrals in the Calculus of Variations. Die Grundlehren der mathematischen Wissenschaften, Band 130. Springer, New York (1966)
Moser, R.: Partial Regularity for Harmonic Maps and Related Problems. World Scientific, Singapore (2005)
Moser, R.: Intrinsic semiharmonic maps. J. Geom. Anal. 21(3), 588–598 (2011)
Nash, J.: The imbedding problem for Riemannian manifolds. Ann. Math. 2(63), 20–63 (1956)
Price, P.: A monotonicity formula for Yang–Mills fields. Manuscr. Math. 43(2–3), 131–166 (1983)
Rivière, T.: Everywhere discontinuous harmonic maps into spheres. Acta Math. 175(2), 197–226 (1995)
Rivière, T.: Conservation laws for conformally invariant variational problems. Invent. Math. 168(1), 1–22 (2007)
Schikorra, A.: Regularity of \(n/2\)-harmonic maps into spheres. J. Differ. Equ. 252(2), 1862–1911 (2012)
Schikorra, A.: \(\varepsilon \)-regularity for systems involving non-local, antisymmetric operators. Calc. Var. Partial Differ. Equ. 54(4), 3531–3570 (2015)
Schoen, R., Uhlenbeck, K.: A regularity theory for harmonic maps. J. Differ. Geom. 17(2), 307–335 (1982)
Schoen, R.M.: Analytic aspects of the harmonic map problem. In: Chern, S.S. (ed.) Seminar on Nonlinear Partial Differential Equations (Berkeley, CA, 1983), Math. Sci. Res. Inst. Publ., vol. 2, pp. 321–358. Springer, New York (1984)
Simon, L.: Theorems on Regularity and Singularity of Energy Minimizing Maps. Springer, Berlin (1996)
Turesson, B.O.: Nonlinear Potential Theory and Weighted Sobolev Spaces, Lecture Notes in Mathematics, vol. 1736. Springer, Berlin (2000)
Acknowledgements
This work is based on the authors PhD thesis, although improvements of the results regarding higher regularity have been made for this article. Much of this work was conducted whilst the author was a postgraduate student at the University of Bath under the supervision of Professor Roger Moser. The author would like to thank Professor Moser for the opportunity to work on this project, for many interesting and helpful discussions about this work and for his unwavering support. The author is also grateful to the referees for their feedback.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M. Struwe.
James Roberts was supported by an Engineering and Physical Sciences Research Council DTA Studentship for much of this research.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Roberts, J. A regularity theory for intrinsic minimising fractional harmonic maps. Calc. Var. 57, 109 (2018). https://doi.org/10.1007/s00526-018-1384-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-018-1384-0