Partial Regularity for Holonomic Minimisers of Quasiconvex Functionals

We prove partial regularity for local minimisers of certain strictly quasiconvex integral functionals, over a class of Sobolev mappings into a compact Riemannian manifold, to which such mappings are said to be holonomically constrained. Our approach uses the lifting of Sobolev mappings to the universal covering space, the connectedness of the covering space, an application of Ekeland’s variational principle and a certain tangential A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{A}}$$\end{document}-harmonic approximation lemma obtained directly via a Lipschitz approximation argument. This allows regularity to be established directly on the level of the gradient. Several applications to variational problems in condensed matter physics with broken symmetries are also discussed, in particular those concerning the superfluidity of liquid helium-3 and nematic liquid crystals.


Introduction and Statement of Main Result
Let M be a compact submanifold of R N without boundary and ⊂ R n be a Lipschitz domain. We define the Sobolev space of mappings into M by Given an integral functional of the form where v : ⊂ R n → R N with n, N 2, we say: Note that the competitor v to the minimiser u must fulfil the manifold constrained condition as well as agreeing with the trace of u on the boundary-in the sense that v − u ∈ W 1, p 0 ( , R N ), where W 1, p 0 ( , R N ) is the closure of the smooth compactly supported functions C ∞ c ( , R N ) in the · W 1, p -norm. In this paper we shall consider an integrand f : × R N × R N n → R, for p 2, which satisfies the following structural assumptions, namely: for some constant > 0 and all (x, u, ξ).
We remark that (h4) is obviously satisfied if f (x, u, ξ) |ξ | p but also allows for integrands with varying sign. It is easy to show that the growth condition (h1) together with the quasiconvexity of (h2) is enough (in fact f need only to be rankone convex) for the derivative D ξ f to satisfy the growth condition and As remarked in [3], there is no growth condition placed on the second order derivatives of ξ → f (x, u, ξ). We merely have, for any given ⊂⊂ and any given > 0, the existence of a finite K such that There are also no global uniform continuity assumptions imposed on D 2 ξξ f (x, u, ξ) either. One only has (u, ξ) → D 2 ξξ f (x, u, ξ) uniformly continuous on compact subsets. In fact, associated to > 0, a min-max construction implies the existence of a non-decreasing concave modulus of continuity ω : [0, +∞) → [0, 1], with lim t 0 ω(t) = 0 and ω(t) = 1 for all t 1, such that for any |u i |, |ξ i | + 1 and x ∈¯ . In considering the above geometric and analytic aspects, we seek to prove the following main result: for some 0 < β < 1.
A non-empty space X is 0-connected if and only if it is pathwise connected. A space X is 1-connected if and only if it is simply connected.
Recall also that (1) every connected manifold admits a universal covering by a simply connected manifold; (2) a connected compact manifold has a compact universal cover if and only if the fundamental group of the base space is finite; and that (3), the homotopy groups of the covering space of order higher than 2 are equal to the homotopy groups of the base space (cf. [49,52]). In our proof, the compactness and connectedness of covering space is needed in Section 4 for the construction of suitable comparison maps. Example 1.3. If k 2 is an integer we have: When the manifold M is a connected compact Lie group G, and g is the Lie algebra of G, the simply connected covering group splits (cf. [57,Theor. 6.6]) into M = R λ × G 1 × · · · × G ν , where λ = dim R c of the centre c of g. As the simply connected simple compact Lie groups G μ are 2-connected we have 2 p < 4. 2 Remark 1.4. From the above theorem and its proof one would expect the Hausdorff dimension of Sing u to be strictly smaller than n when u is additionally assumed to be Lipschitz continuous (cf. [59] for unconstrained Lipschizian minimisers of strictly quasiconvex integral functionals with p-growth).
The canonical holonomically constrained variational problem is that of harmonic maps between Riemannian manifolds which minimise the Dirichlet energy. The question of regularity for minimisers of the Dirichlet energy in this constrained context was first obtained by Morrey [70] in connection with his analysis of Plateau's problem. By working with absolutely continuous representatives Morrey showed minimisers to be locally Hölder continuous when n = 2. The harmonic map regularity result was extended to the case n > 2 by the seminal work of Schoen and Uhlenbeck [79]. By working with the class (1) they showed energy minimising maps with a compact image to be locally smooth away from a closed (singular) subset that is discrete if n = 3 and has Hausdorff dimension at most n −3 when n 4. A refined treatment of this partial regularity result, as well as further analysis on the rectifiability of the singular set, can be found in the book of Simon [78]. Similar results for the case in which the image lies in a single coordinate chart were obtained by Giaquinta and Giusti [40,41] (and by [54,58] when the image of the map is contained in a convex ball of M). In the p-harmonic case, Hardt and Lin [56] showed p-energy minimisers, for p > 1, to be C 1,α loc for some 0 < α < 1 except possibly on a closed set of Hausdorff dimension at most n − p − 1. Further partial regularity results in the p-harmonic case for maps into a compact target manifold can be found in [28,64]. The question of minimising in an appropriate homotopy class has also been studied by White [83,84].
Partial regularity results for a more general class of convex functionals have been obtained by Luckhaus [63] with appropriately well-behaved blow-ups. In this direction, results for convex functionals with p-growth can also be found in [53,56]. In particular, [56] obtained reverse Hölder inequalities under the assumption M is ( p − 1)-connected via a retraction of R N \X onto M for some polyhedron X in R N \M. This kind of retraction was previously used in (the domain) by [83] for the existence of finite energy extensions of Lipschitz maps. In fact the idea of using such a continuous retraction with analytic estimates goes back to the proof in [37] of isoperimetric inequalities.
To prove the existence of (unconstrained) minimisers of general integral functionals subject to given boundary data, Morrey [71] introduced a notion of quasiconvexity which isolated a property of the integrand that, under appropriate growth conditions, is necessary and sufficient for the weak sequential lower semicontinuity of the functional. A result of Acerbi and Fusco [2] shows that a non-negative Carathéodory integrand with p-growth is weakly sequentially lower semicontinuous on the W 1, p -Sobolev space if and only if the integrand is quasiconvex. Regularity considerations for such quasiconvex functionals in the unconstrained case were initiated by Evans [32], who showed a slightly strengthened version of quasiconvexity (that is the so-called uniformly strict quasiconvex condition) forces partial regularity. That is to say, if u minimises a functional of the form´ f (Du) dx, where f is a strictly quasiconvex C 2 -function with p-growth, p 2, that satisfies an additional bound on the second derivatives D 2 f , then the gradient Du is locally Hölder continuous outside a set of Lebesgue measure zero. This result was subsequently generalised to integrals with x-and u-dependences in [39,46] and the bound for D 2 f was removed in [3]. An extension to the subquadratic case 1 < p < 2 has also been obtained in [17]. A partial regularity result for a class of strong local minimisers of quasiconvex functionals can be found in [60]. Further details on other aspects in the unconstrained case [including boundary regularity, degenerate cases, conditions for everywhere regularity, ω-minima, ( p, q)-growth conditions and counterexamples to full regularity] can be found in the survey article [69].
In order to obtain partial regularity, a linearisation strategy is often used to transfer the 'good' regularity properties of the linearised elliptic system to the nonlinear problem. The technique of harmonic-type approximations, in particular, has proved both a popular and elemental way of obtaining partial regularity for solutions to non-linear elliptic systems and minimisers of certain integral functionals (cf. [29]). The underlying premise of this method goes back to ideas introduced by De Giorgi [22] in his treatment of minimal surfaces. Specifically, he establishes 'some properties of regular surfaces approximating minimal surfaces; these properties will be obtained by a comparison argument with the graph of suitable harmonic functions' [23, p. 244]. In the context of regularity for elliptic systems the technique was utilised by Simon [78, §1.6] to provide a neat and elementary proof of partial regularity. For quasiconvex integrals this method was initiated in [24]. The proof of harmonic-type approximation results is typically achieved through an indirect compactness argument, nonetheless a version using 'a direct approach based on the Lipschitz truncation technique which requires no contradiction argument' has recently been proved in [27,30].
For minimisers of quasiconvex functionals in the manifold constrained case, the only know result to date (aside from the results in [53,56,63] for holonomic minimiser of convex functionals) is that of the Evans and Gariepy paper [34]. By appealing to the blow-up technique introduced in [33], Evans and Gariepy consider a functional of the form´ f (Du) dx under the structural hypothesis that f is a strictly quasiconvex C 2 -function for p = 2 which satisfies an additional bound on the second derivatives D 2 f . Comparison maps satisfying the non-linear manifold constraints in the indirect blow-up proof are obtained from a particular extension lemma of Luckhaus (cf. [63,64] and also [48,55,78]), however this construct only works in the blow-up case due to the order in which limits are taken (cf. [34, p. 87]).

Remark 1.5. (Idea of proof and applicability of Luckhaus' lemma)
where M m ⊂ R is a Riemannian manifold without boundary and N ⊂ R k is a closed subset of positive reach with a neighbourhood O, the so-called Luckhaus lemma seeks to construct a function f ∈ W 1, p (M × (0, λ), R k ) for 0 < λ < 1, with f (·, 0) = f 1 and f (·, λ) = f 2 , such that one has L p -control of the dist( f, N )-so that it can be made to be sufficiently small. Normally in the unconstrained setting f would be constructed simply as a convex combination of f 1 and f 2 , however to gain the extra control on dist( f, N ) a triangulation of the domain M is employed so as to restrict the Sobolev mappings to lower-dimensional skeletons. For such skeletons of dimension d p − 1 one can use a convex combination and a Morrey-type embedding to construct and gain the desired pointwise control, that is if 1 2 . Then for skeletons of dimension d > p−1 , where the embeddings no longer hold, a homogeneous degree 0 extension is used to inductively extend the lower-dimensional construction. The result of this construction on the triangulated domain yields an f ∈ W 1, p (M × (0, λ), R k ) such that´M ×(0,λ) |∇ f | p CλA and ess sup Now in the application of Luckhaus' lemma to regularity proofs, the need for the distance to be uniformly small is directly juxtapose with the control of dist( f, N ) being derived from Sobolev embeddings (as outlined in the above). For the Dirichlet energy the conflict between these requirements is resolved by appealing to a monotonicity formula (cf. [78]). Indeed, for the class of convex integrands considered in [63] a monotonicity formula is also needed. In the f (x, u, Du)-situation considered here, higher integrability is required in order to carry out Ekeland's variational principle (note that the indirect approach of [34] no longer works with the u-dependence). In this regard the approach of [56], using connectedness assumptions on the target manifold, dispenses with the need for a monotonicity formula and allows for the application of Gehring's lemma.
The above Theorem 1.2 considers holonomic minimisers of quasiconvex integrals which also have x-and u-dependences. Here we focus on the case p < n (for if p n the Sobolev embeddings imply the regularity argument would follow the single coordinate chart case). Our proof of Theorem 1.2 is direct via an application of Ekeland's variational principle; a certain tangential A-harmonic approximate lemma; Caccioppoli inequalities using only minimality, the p-growth of (h1) and the strong quasiconvexity of (h2); the Sobolev lifting result of Theorem 4.1 below; and an extension-type result using the -connectedness of the compact covering space M. The resulting estimates allow a sufficient decay of the excess functions E given by where V (·) is given by (9) below. 3 Regularity follows via Campanato's integral characterisation of Hölder continuity [16, teor. I.2] with the singular set characterised by Remark 1.6. If we take a point x ∈ \Sing u it follows that lim sup |(u) x,r | < +∞, lim sup |(Du) x,r | < +∞ and lim inf E(x, r ) = 0 by definition. Then there exists > 0 and 0 < R < dist(x, ∂ ) such that |(u) x,r | < and |(Du) x,r | < for all 0 < r < R. Similarly, for every ε > 0 there exists 0 < R < dist(x, ∂ ) such that E(x, R) < ε. Now if for every > 0 we can find ε = ε( ) > 0 and 0 < R = R( ) < dist(x, ∂ ) both sufficiently small such that . Thus we need to show, for some 0 < β < 1 and every x ∈ ,ε,R , that 3 We denote the average (h) = ffl h = 1 | |´ h for any h ∈ L 1 ( , R N ), where | | denotes the Lebesgue measure of any ⊂ R n , and use the abbreviation (h) x,r , or simply (h) r , for the mean value of h on an open ball B r (x) ⊂ R n of radius r centred at x. for some K = K (β, ε, , R) > 0, in order to conclude Du ∈ C 0,β loc ( ,ε,R , R N n ) by Campanato's integral characterisation of Hölder continuity. Consequently Du is C 0,β loc -continuous on of which Sing u is the complement in , provided one has (7) and can establish the excess-decay (8).
In the upcoming analysis we make use of the following auxiliary function for every ξ ∈ R k . Note that R + t → |V (t)| is convex and |ξ | 2 , |ξ | p |V (ξ )| 2 c( p)(|ξ | 2 + |ξ | p ) when p 2. Also from the proof in [25, §4] we have, for any for some constant c = c(n, N , p) > 0, where B R ⊂ R n , n 2 and 2 * = 2n n+2 < 2. The following properties of (9) will also be called upon. . Let p 2 and V : R k → R k be as in (9). Then for any ξ, η ∈ R k and t > 0 there holds: (11f)

Applications to Problems in Condensed Matter Physics
The manifold constrained regularity problem has several applications to problems in condensed matter physics. For such applications the (unconstrained) mapping is a tensor order parameter T = (T αβ ) : ⊂ R n → R N with an integral functional of the formˆ The integrand is taken to be invariant under a given symmetry group G and is typically expanded as power series with respect to this symmetry group. Critical points of the bulk energy f B form an orbit of solutions corresponding to a subgroup H ⊂ G of residual symmetries. The resulting manifold of internal states M = G/H describes the 'broken symmetries' of the system; such T taking values in this group orbit (for which the bulk energy f B remains constant) form the corresponding constrained mapping case. Example 1.8. (Nematic liquid crystals) In the Landau-de Gennes theory for nematic liquid crystals the state of alignment is characterised by a tensor order parameter Q as a traceless symmetric 3 × 3 real matrix, that is In a general nematic phase the order parameter has five degrees of freedom (two specifying the degree order with the other three for the principal directions). The symmetry group is G = SO(3) with the group action defined by conjugation for any R ∈ SO(3). The Landau-de Gennes free-energy functional (cf. [20]), taking into consideration spatial variations of the local order parameter, is of the form The bulk free energy f B is invariant under the SO(3)-action by conjugation on the five-dimensional space of Q-tensors; its expansion to fourth order takes the form where a, b and c are material and temperature dependent constants (cf. [66]). The expansion of the free energy f grad in SO(3)-invariant powers of Q i j , and its derivatives up to the order Q Q ∇ Q ∇ Q, contains twenty-two terms and four surface relations (cf. [11,65,81]). For instance the lowest order terms of the form with the corresponding surface relation  30 bar is applied-under less pressure the small atomic mass and the weakness of the attractive interactions prevents the formation of a crystalline solid with a rigid lattice structure. This is the first indication of the remarkable macroscopic quantum effects in these systems. Quantum effects are also responsible for the strikingly different behaviours of 3 He (obeying Fermi-Dirac statistics) and 4 He (obeying Bose-Einstein statistics) at lower temperatures: the superfluid transition temperature of helium-4 is 2.17 K at 1 atm, whereas for helium-3 it is below 3 mK and in fact forms several superfluid phases. 5 In the case of superfluid helium-3, the order parameter for a phase φ with rotation matrices R (1) and R (2) in the respective spin space S and orbital space L. In regions near the critical temperate T c (that is the Ginzburg-Landau regime) the bulk free energy f B can be obtained from A μj by contracting all indices in second and fourth order terms that are SO The resulting expansion yields where the coefficients α and β i depend on external parameters like temperature and pressure (cf. [74]). When allowing for spatial variations in the order parameter, the gradient energy of an anisotropic superfluid in the Ginzburg-Landau regime has leading SO(3) × SO(3) × U (1)-invariant terms quadratic in the gradient of the form  3 He-A Further information on broken symmetries with regards to gradient, dipole and magnetic interactions can be found in [67,80,82] where γ i are given constants (cf. [1,21]). Note that the terms are dependent in the sense that the second term can be written in terms of the third plus a divergence terms which vanishes under appropriate boundary conditions. The unusually large number of internal degrees of freedom allow for a rich array of superfluid phases of liquid 3 He. Fortunately it is possible to formulate a systematic approach to the investigation of all possible symmetry breakings; the classification of all continuous and discrete subgroups of SO(3) × SO(3) × U (1) can be found in [15]. It turns out that the continuous subgroup with the highest remaining symmetry corresponds to the so-called B-phase of [76] that is stable down to the lowest temperatures attained so far. In this phase the order parameter is invariant under joint rotations of L and S represented by the subgroup SO(3) L+S . The corresponding broken symmetries On the other hand, if one is confined to a finite temperature range and is above a critical pressure of about 21 bar, one enters a different phases: the so-called A-phase of [76]. In this dipolefree A-phase the subgroup of residual symmetries is The corresponding space of broken symmetries is the five dimensional manifold Other manifolds of broken symmetries can be considered if an external magnetic field is also considered (cf. Table 1). For a review of the theory of anisotropic superfluid phases of helium-3 see [62].

Tubular Neigbourhoods
For an embedded submanifold M in R N without boundary, and its normal bundle π : As the differential E * is surjective along this zero section, E is a local diffeomorphism by the inverse function theorem. One then has the following existence result: is called the retraction onto M.
Proof. The existence of O follows from Theorem 2.1. As the normal bundle π : where v is normal to M at x, and the differential E * | (x,v) is a critical point with nullity ν > 0; they are points where nearby normals intersect. By [68, Lem. 6.3] these focal points are precisely of the form x + κ −1 i v, where κ i are the principal curvatures of M at x in the unit normal direction v.
. For instance, one could take K to be the closure of the tubular neighbourhood {x + v : x ∈ M, v ∈ (T x M) ⊥ , |v| < ε}, for ε > 0 sufficiently small, so as to remain 'strictly away from focal points'. Furthermore, each ∇ ( j) are uniformly bounded and uniformly continuous on compact sets for j = 0, . . . , r − 1.
Note that the boundary of a manifold cannot have a tubular neighbourhood strictly in the above sense. However, there is a retraction of an open neighbourhood onto ∂M. This is obviously possible locally, and two local retractions into coordinate domains can be glued together with a bump function.

Sets of Positive Reach and the Regularity of the Squared Distance Function
For a subset A ⊂ R N let the distance function δ : R N → R be defined by Furthermore, let Unp(A) be the set of all points x ∈ R N for which there exists a unique point of A nearest to x and, like in the previous section, let the map : If a ∈ A then reach(A, a) is the supremum of the set of all numbers r for which {x : |x − a| < r } ⊂ Unp(A). We then define reach(A) = inf{reach(A, a) : a ∈ A}. If reach(A) > 0 then it is said that A has positive reach. A well known characterisation of convexity (going back to independent results by Bunt, Kritikos and Motzkin, cf. [19]) states that a subset A ⊂ R N has reach(A) = +∞ if and only if A is closed and convex.
By introducing the squared distance function η(x) = 1 2 δ(x) 2 we have: Theorem 2.4. [36, 4.8] For every nonempty closed subset A of R N the following statements hold with U = Unp(A): For a compact C r -submanifold M ⊂ R N without a boundary, the distance function δ is C r −1 near M (since it is continuous and can be written in terms of directions normal to M); in fact the squared distance function η ∈ C r (U ) for r 2 (cf. [38], [47, p. 354]). Furthermore, by approximating M locally at a point x as the graph of a quadratic polynomial given by the second fundamental form, one can show for any z ∈ R N with 0 < |z| 1 sufficiently small, using the elementary geometry of the normal line at (x + z) through x + z and the normal line at x along the projection of z onto (T x M) ⊥ , that where the matrix Q represents the orthogonal projection onto the normal space of M at the point x. On the other hand, the Taylor series of η is η( , since η(x) = 0 and ∇η(x) = 0 due to the projection (x) = x. Therefore ∇ 2 η represents the orthogonal projection onto the normal space of M and P = I − ∇ 2 η is the projection operator onto the tangent space. Moreover, by taking a further derivative of (4) in Theorem 2.4 we conclude: Example 2.6. For the n-sphere S n ⊂ R n+1 the retraction : x → x |x| for x = 0. Thus one finds that P(u) = ∇ (u) = I − u ⊗ u at points u ∈ S n , where I is the identity matrix.

Constrained Stationary Variations
Consider the functional For a compact C 3 -Riemannian submanifold M ⊂ R N without boundary, we derive the Euler-Lagrange equations for holonomically constrained minimisers, using the retraction (12), as follows: for some constant C > 0 and all ξ ∈ R N n . Then for any holonomic F-minimiser Example 3.2. When considering the Dirichlet energy´ |Du| 2 dx the above lemma implies holonomically constrained minimisers satisfŷ One then concludes  (13) are said to be weakly harmonic maps.

A Construction of Suitable Comparison Maps
In seeking to prove the regularity of vector-valued maps which minimise an integral functional subject to a family of smooth nonlinear (manifold) constraints, a major technical difficulty resides in finding comparison maps which satisfy the manifold constraints (particularly since one cannot localise the problem in the image for n > 2). In the case of harmonic maps into compact manifolds, one can construct comparison maps via a lemma of Luckhaus originally proved in [63, Lem. 1] and revised in [64,Lem. 3] (which develops an earlier extension lemma of [79,Lem. 4.3]). Using this and a monotonicity formula for harmonic maps it is possible to prove Caccioppoli inequalities near points with small scale energy (cf. [78, §2.8]). However, without such a monotonicity formula available in general one needs to construct comparison maps absent of any need for initial 'smallness assumptions'. Our approach here is to combine the lifting of Sobolev maps to the universal covering together with an extension result using the connectedness of the covering space.

The Lifting of Sobolev Mappings
For a given connected compact Riemannian manifold M without boundary and an open simply connected subset ⊂ R n we ask if any mapping u ∈ W s, p ( , M) factors through the universal covering π : M → M as regular as u permits, that is if can be completed to a commutative diagram with ϕ ∈ W s, p ( , M). Of course if M is simply connected then M = M and π is the identity map. Hence we are concerned with the non-trivial case π 1 (M) = 0.
For such a problem we have the following affirmative result when 2 p ∞ and s = 1.

A Locally Lipschitz Retraction
We recall some basic notions pertaining to the triangulation of a manifold. Firstly, we say a cell is a bounded subset of R m consisting of all points x satisfying a system of finite linear inequalities i (x) = a i j x j b i ; it is a compact convex subset of R m . The dimension of is the dimension of the smallest dimensional plane containing . If is an m-cell, ∂ (the boundary of ) is the union of a finite number of (m − 1)-cells, each the intersection of an (m − 1)-plane with . Each of the (m − 1)-cells i into which ∂ is decomposed is called a face of ; each of the (m − 2)-faces of i is called an (m − 2)-face of . As is given by a system of linear inequalities, replacing an inequality by equality determines a face of and conversely.

Definition 4.2.
A cell complex K is a collection of cells such that each face of a cell in K is also in K , the intersection of two cells in K is either empty or a common face, and each point of the polyhedron (as the union of all cells of K ) has a neighbourhood intersecting finitely many cells of K . The dimension of K is the maximal dimension of the cells in K .
A subcomplex K is a subset of K which itself is a complex; to verify a subcomplex is indeed a complex it is clear that one only needs to check that faces of each cell in K are in K . In particular, the k-skeleton K k is the subcomplex of K consisting of all cells of dimension at most k. In the case when the cell is a cube in R m centred at the origin with faces parallel to the coordinate axes, the dual skeleton of dimension j is the set of points in the cube which have at least m − j components equal to zero. By using an isometry, the dual skeleton of a cube in R m of dimension m can be defined in a general position. Thus the j-dimensional dual skeleton L j of a complex K of cubes is simply the union of the dual skeletons of dimension j of each cube in the complex (cf. [12, §2.1]).
A Lipschitz triangulation f : K → M of a given manifold M can be such that the complex K (given by Definition 4.2) consists of cubes each of which is isometric to the standard cube [−1, 1] m ⊂ R m (that is K is a 'cubication' of M). 6 A result by Whitehead [85,Theor. 7] states that any compact C 1 -manifold without boundary has a triangulation in the C 1 -class. This was extended by Munkres [72, Chap. 2] to include C r -manifolds with boundary for 1 r ∞. In particular: Theorem 4.3. Every compact C r -manifold without boundary has a C r -triangulation. If M is a manifold having boundary, any C r -triangulation of the boundary may be extended to a C r -triangulation of M.
Observe also that |Dφ (x)| C/|x| . Given this, the same construction can be performed for every cube in K . If two cubes have a non-empty intersection the corresponding maps φ ,1 and φ ,2 coincide on the common face. Hence we can glue together the locally Lipschitz continuous maps obtained for each of the cubes in the complex K so as to obtain a global map ψ with the desired properties.
for some constant C > 0 depending on k, and M.
This result was first proved in [56, Lem. 6.1] using several results from homotopy theory (see also [13,Lem. 2.2]). Our proof below is based on the elementary result, for a Riemannian manifold Y , that This extension result follows from the fact that continuous mappings and homotopies can be approximated by smooth ones (cf. [61,Theor. 6.26]), however if the smooth manifold Y is replaced by a metric space the result no longer holds (cf. [26, §4]). Proceeding inductively, suppose we now have a Lipschitz map K j ∪ T m → M for the j-dimensional skeleton K j with integer 2 j . Then for any ( j + 1)-simplex j+1 with at least one j-dimensional face or lower contained in the complex T m of M such that the simplex is not contained in T m , we can use the induction hypothesis, ∂ j+1 S j and the fact that π 0 (M) = · · · = π j−1 (M) = 0 to obtain a Lipschitz map φ j : S j → M. As π j (M) = 0 it follows that can be completed to a commutative diagram by Proposition 4.6. Hence we can map j+1 → M for all ( j + 1)-simplexes j+1 that meet T m with at least one j-dimensional face or lower contained in T m . Then as before, for any ( j + 1)simplexes j+1 that do not meet T m at all, we can extend ∂ j+1 → M into the interior by ' j-face extensions'. Hence we have a Lipschitz map K j+1 ∪ T m → M for the ( j + 1)-dimensional complex K j+1 . As the obstruction of π +1 (M) = 0, we then have a Lipschitz retraction by the above construction. Furthermore, by taking a 'cubication' subcomplex K of K if necessary, Lemma 4.4 yields a locally Lipschitz retraction

An Extension Result for Sobolev Mappings
From the trace map R :  [7] has shown the condition of -connectedness of M for 1 < p < + 2 (with p < n) is both necessary and sufficient for the Sobolev extension problem. For the purposes here we appeal to a construction in [53, Lem. A.1] (when the target manifold is S 2 ), and the generalisation given in [56,Lem. 6.2], to prove the following:  M). Then there exists ψ ∈ (φ + W Also, by a change of variables and the definition of the dual skeleton, there exists a positive constant C = C( p, k, , M) such that since dim X k − − 2 and 1 < p < + 2. Now for a sufficiently small 0 < σ < min{ ρ 2 , 1 2 dist( M, ∂ Q)} and a point a ∈ B k σ = {ξ ∈ R k : |ξ | < σ}, denote the translations Q a = {ξ + a : ξ ∈ Q} and X a = {ξ + a : ξ ∈ X } so that one has the retraction P a : Q a \X a → M given by P a (ξ ) = P(ξ − a). Then by Fubini's theorem and (15) there exists some Cˆ |Dφ| p and as |ξ | 2 , |ξ | p |V (ξ )| 2 c(|ξ | 2 + |ξ | p ), since p 2, we also havê for almost everywhere x ∈ . Furthermore, for some K > 0 we havê for a 'good set' of a ∈ B k σ (if not then Markov's inequality implies the inequalities fails with an exceptional set less than or equal to K −1 |B k σ |). Therefore there exists an a 0 with |a 0 | σ such that for almost everywhere x ∈ . By the chain rule and (14) we have Lip(P a | M ) C, for some C > 0 independent of a, provided σ is sufficiently small. Furthermore, as the dimensions of the domain and codomain for P a | M are equal and P| M is equal to the identity map on M, the inverse function theorem implies that sup a∈B k σ Lip(P a | M ) −1 is finite for sufficiently small σ . By setting the desired conclusion follows.

Remark 4.8.
One can consider the model case for the above construction by setting In this spherical target case the above construction can in fact be explicitly carried out as follows (cf. [53]). Firstly, let P a : R k \{a} → S k−1 send ξ → ξ −a |ξ −a| . As the gradient provided 1 < p < k. Therefore, by Markov's inequality there exists an a 0 with |a 0 | 1 2 such thatˆ Now consider the map (P a | S k−1 ) −1 : S k−1 → S k−1 , for any |a| 1 2 , which sends Since |D(P a | S k−1 ) −1 (η)| C uniformly independent of a, the desired conclusion follows by setting ψ = (P a 0 | S k−1 ) −1 • P a 0 • φ ∈ W 1, p ( , S k−1 ).

A Hole-Filling Comparison Map Construction
We can construct the following suitable comparison maps by appealing to the lifting result of Theorem 4.1 together with the extension result of Lemma 4.7. Note that the connectedness assumption on the covering space is deemed canonical as a covering space is a universal covering space if it is simply connected.

Higher Integrability
Given the results of Section 4 we obtain Caccioppoli inequalities of the first kind using only the growth (h1), (3) and the coercivity (h4). An application of Gehring's lemma then implies local higher integrability for the gradient Du.
From (3) we estimatê p (B s , M). Now by the lifting result of Theorem 4.1 there exists ϕ ∈ W 1, p (B s , M) such that u = π • ϕ with |Dϕ| = |Du| almost everywhere. Then by Lemma 4.9 one has a suitable competitor v ∈ (u + W 1, p (B s , R N ) So by Widman's hole filling trick we havê  We can also get higher integrability up to the boundary by adapting the proof of the previous lemma using standard up to the boundary arguments (cf. [5,Lem. 3.5], [42, p. 152]) together with a suitable up to the boundary version of Lemma 4.9. p (B, M), then there exists C > 0, q ∈ ( p, q 0 ] such that u ∈ W 1,q (B, M) and

An Application of Ekeland's Variational Principle
In this section we use the following lemma of Ekeland together with ideas from Acerbi and Fusco [3] to show an holonomic F-minimiser almost minimises the so-called frozen problem. This result is then subsequently utilised in Section 8 to show approximate tangential harmonicity via a linearisation strategy applied to the frozen integrand. Lemma 6.1. [31] Let (X, d) be a complete metric space and J : X → R ∪ {+∞} be a lower semicontinuous functional that is bounded from below but not identically +∞. For ε > 0 and a point u ∈ X such that To satisfy the lower semicontinuity requirement we recall the following: is sequentially weakly lower semicontinuous on (u+W Now for a given u ∈ W 1, p loc ( , M) and a fixed ball is a minimiser of the the corresponding frozen problem: for all w ∈ (u+W Note that the existence of u is obtained by direct methods whereby the Rellich-Kondrachov theorem assures the limit u of the minimising subsequence satisfies the condition u ∈ M almost everywhere.
By defining we see (following the proof of [3, Lem. IV.7] together with Lemmas 5.1, 5.2, (h3) and the minimality of u) that any holonomic F-minimiser u of the 'full problem' is an almost minimiser of the 'frozen problem' in the following sense:  (20).  (21) which is also a minimiser in the sense that Moreover, by following the proof of Lemma 3.1 we also obtain the following: where is a retraction from a tubular neighbourhood onto M, provided the integrand f satisfies (h1).

The Tangential A-Harmonic Approximation Lemma
A symmetric bilinear form A on R N n is said to be strongly elliptic in the sense of Legendre-Hadamard, or simply elliptic, if for all η ∈ R N and ξ ∈ R n there holds for some > 0. We call an ellipticity constant associated to A. Note that A is bounded [in the sense that A(ζ 1 , ζ 2 ) L|ζ 1 ||ζ 2 | for all ζ 1 , ζ 2 ∈ R N n ] for free (by finite dimensionality). A classical result shows this condition to be equivalent to the strong ellipticity condition for all φ ∈ W 1,2 0 ( , R N ). Looking to the notion of harmonicity used in the limit of indirect blow-up proof of [34, pp. 83-5] we say: Here is the retraction given by (12) and ∇ (θ) is a projection onto the tangent space T θ M (cf. Corollary 2.5). Hence the condition ∇ (θ)Dh = Dh is equivalent to requiring D i h ∈ T θ M for i = 1, . . . , n. Furthermore, we also introduce the map C θ : R N → θ + T θ M that sends M) is a δ-approximate tangential Aharmonic map about a given point θ ∈ M with exponents p 2, α > 1, 0 < β 1 if A satisfies the strong ellipticity condition (23) and for some δ > 0 we have The following a priori estimates for tangential A-harmonic maps are obtained by a change of orthonormal basis. This reduces the problem to an unconstrained elliptic system from which the result can be obtained. .
weakly with´ B(Dϕ, Dϕ) ´ |Dϕ| 2 for all ϕ ∈ C 1 c ( , R m ). In which case H is a solution to an (unconstrained) linear elliptic system with constant coefficients satisfying the strong ellipticity condition (23). Hence H ∈ C ∞ loc with the desired interior estimate obtained from [17, pp. 150-1]. Remark 7.4. As |Dh − (Dh) r | 2r sup B r |D 2 h| on B r , the above lemma implies |Dh − (Dh) r | C ffl B 2R | r R Dh| for 0 < r < R. Since |V (·)| 2 is a convex function for p 2 it follows that By appealing to the same change of variables argument used in Lemma 7.3 and the L p -estimates of Campanato and Stampacchia (cf. [18], [44, §7.1.1]) we also obtain the following: such that ∇ (θ)Dφ = Dφ and ∇ (θ)Dv = Dv almost everywhere, where A is strongly elliptic bilinear form of constant coefficients, then Dv ∈ L p ( , R N n ) and for some c > 0 independent of v and F, provided ⊂ R n is a smooth bounded domain.
Let us also recall the following standard pointwise approximation of Sobolev functions that goes back to a result by [4]. |{v λ = v}| cλ − pˆ |v| p + |Dv| p for some c = c(n, p) > 0.
We can now give a (direct) proof of the following tangential A-harmonic approximation lemma based on the the general Sobolev-Orlicz space proof of [27,30].  = δ(ε, n, N , p, α, , L) , is any δ-approximate tangential A-harmonic map about a point θ ∈ M with exponents α > 1 and 0 < β 1, where A is a strongly elliptic bilinear form with ellipticity constant and an upper bound L, then the unique tangential A-harmonic map h ∈ C θ (u) + W Note that the existence and uniqueness of the tangential A-harmonic map follows from the standard linear theory.
for any ϕ ∈ W 1, p 0 (B R , R N ) such that ∇ (θ)Dϕ = Dϕ and ∇ (θ)Dv = Dv almost everywhere. By Lemma 7.5 we havê N , q, , L). From this L q -estimate, Equation (26) and the fact that U ∈ W 1, p since h is a tangential A-harmonic map about θ . Similarly, the strong ellipticity condition (23) also implies since h is a tangential A-harmonic map about θ with U ∈ W 1,2 0 (B R , R N ) also. Now set A = (Du) R and denote q = ffl B R |Dv| q . By Lemma 7.6 there exists a Lipschitz approximation v λ ∈ W 1,∞ 0 (B R , R N ) of the above v with λ ∈ (k , k q ) for some integer k > 1. In which case we can write Using Lemma 7.6 and α > 1 we bound the first term N , p, , L). Again, using the estimates from Lemma 7.6 we also bound N , p). Thus for every 0 < ε < 1 we may choose k = k(ε, α, n, N , p, , L) > 0 sufficiently large so that As u is a δ-approximate tangential A-harmonic map about θ , Definition 7.2 implies, together with the bound Dv λ ∞ λ k q , Young's inequality and p 2, that the second term Furthermore, we also have for some positive c 3 = c 3 (n, N , p, , L), where by Definition 7.2 we have By choosing sufficiently small δ = O(ε 2+ 1 α−1 ), that is a δ satisfying δ K ε 2+ 1 α−1 (28) for some K = K (n, N , p, α, , L) > 0, we can bound 2 If we denote 2 = ffl B R |DU | 2 , recalling U ∈ W 1,2 0 (B R , R N ) from the above, then by Lemma 7.6 there exists a Lipschitz approximation U μ ∈ W 1,∞ 0 (B R , R N ) to U with μ ∈ ( j , j 2 ) for some integer j > 1. As the δ-approximate tangential A-harmonic condition about θ yields we can, like in the above estimates, choose an integer j = j (ε, α) > 0 sufficiently large and δ = O(ε 2+ 1 α−1 ) sufficiently small such that The desired result now follows by combining the initial estimate (25) with both the L 2 and L p -estimates and (27) for δ = O(ε 2+ 1 α−1 ).

Approximate Tangential Harmonicity
In this section we establish the approximate harmonicity of holonomic F-minimisers u ∈ W 1, p loc ( , M) as per Definition 7.2. To do so we first note that the strong quasiconvexity of (h2) implies u 0 , ξ). By sending ε → 0 the latter inequality becomeŝ So if, for any given > 0, we have |(u) x 0 ,R | and |(Du) x 0 ,R | on some B R (x 0 ) ⊂⊂ , the bilinear form is then both bounded, with an upper bound L = K from (4), and strongly elliptic in the sense of (23) with an ellipticity constant = 2γ . Given this one can use a linearisation strategy to prove the below approximate harmonicity result of Proposition 8.3 via Ekeland's variational principle and higher integrability. The linearisation strategy is applied first to the frozen problem yielding the following: , M) and some 0 < β < 1, then for every > 0 there exists C = C(n, N , p, , K ) > 0 such that into the parts and where (2) and (4), via (31), imply Moreover, on B R ∩ {|Dv − A| 1} Equation (32) and the modulus of continuity (5) imply |G| c ω(|Dv − A|)|Dv − A||Dφ|. Hence by (11a) and (11f) we have where the second last inequality follows by the fact that ω is concave and at most 1 and the last inequality follows by Cauchy-Schwarz and the fact that ω is nondecreasing.
Expanding I I yields By the growth of (2), the bound φ ∞ cR Dφ ∞ and the Lipschitz properties of the retraction (cf. Remark 2.3) we bound Finally, by Corollary 6.4 and the bound φ ∞ cR Dφ ∞ we get Then for every  > 0 and α > 1 there exists c = c(n, N , p, α, ) since Du = D( (u)) = ∇ (u)Du. For any α > 1 we can, using (10), (11a), (11c) and the Lipschitz properties of the retraction (cf. Remark 2.3), bound Dv) x 0 ,R by the divergence theorem. So by (11a) and (11e) we find whenever |(Du) x 0 ,R | . Given this we can, using the fact that the minimiser v of the frozen problem is close to the holonomic F-minimiser u in the sense of (21) via Ekeland's variational principle, establish the following approximate tangential harmonicity result.  N , p, ) sufficiently small, then for any α > 1 there exists C = C(n, N , p, , K , α) > 0 such that Proof. Using the initial assumptions, the result of Section 6 yields a minimiser v ∈ (u + W (22) which also satisfies (21). Furthermore, from (21) and the initial assumptions, the Poincaré inequality implies (21), the Poincaré inequality and the initial assumptions. With θ u = ((u) R ), Then using (4), (5) and the Lipschitz properties of the retraction (cf. Remark 2.3) we bound The result now follows from Lemmas 8.1, 8.2 and the Poincaré inequality, since E(R; v) 2 E(R; u) 2 + C R β by (33) together with (21).

Reverse Hölder-Type Inequalities
We establish reverse Hölder-type inequalities following a Caccioppoli inequality of the second kind argument using strict quasiconvexity and minimality via the comparison map construction from Section 4.  R 1 (n, N , p, ) such that for any B R (x 0 ) ⊂⊂ with 0 < R < R 1 , any u 0 ∈ R N with |u 0 | + 1 and any A ∈ R N n with |A| .
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