# Partial regularity for manifold constrained *p*(*x*)-harmonic maps

## Abstract

We prove that manifold constrained *p*(*x*)-harmonic maps are locally \(C^{1,\beta _{0}}\)-regular outside a set of zero *n*-dimensional Lebesgue’s measure, for some \(\beta _{0} \in (0,1)\). We also provide an estimate from above of the Hausdorff dimension of the singular set.

## Mathematics Subject Classification

49N60 35J50 58E20## 1 Introduction

*p*(

*x*)-harmonic maps. More precisely, we consider local minimizers of the functional

*m*-dimensional, compact submanifold endowed with a suitable topology. We refer to Sect. 1 for the precise notation. Our final outcome is that there exists a relatively open set \(\Omega _{0}\subset \Omega \) of full

*n*-dimensional Lebesgue measure such that \(u \in C^{1,\beta _{0}}_{\mathrm {loc}}(\Omega _{0},\mathcal {M})\) for some \(\beta _{0}\in (0,1)\) and \(\Sigma _{0}(u):=\Omega \setminus \Omega _{0}\) has Hausdorff dimension at the most equal to \(n-\gamma _{1}\). Moreover, after imposing some extra restrictions on the variable exponent \(p(\cdot )\), we are able to provide a further reduction to the Hausdorff dimension of the singular set of \(\mathcal {M}\)-constrained minimizers of the \(p(\cdot )\)-energy

*p*-growth case, both for scalar and vector valued minimizers. Subsequently, the regularity theory of functionals with variable growth has been developed in a series of interesting papers by Ragusa, Tachikawa and Usuba [40, 41, 42, 46, 47], where the authors established partial regularity results for unconstrained minimizers that are on the other hand obviously related to the constrained case. Especially, in [46] Tachikawa gives an interesting partial regularity result and some singular set estimates for a class of functionals related to the constrained minimization problem in which minimizers are assumed to take values in a single chart. This generalizes the well-known results of Giaquinta and Giusti [22] valid in the case of quadratic functionals with special structure. In this paper we finally tackle the case of local minimizers with values into a manifold, provided that suitable topological assumptions are considered on the manifold \(\mathcal {M}\) and optimal regularity conditions are in force on \(p(\cdot )\) and \(k(\cdot )\). Our first main result is the following:

### Theorem 1

Let \(u \in W^{1,p(\cdot )}(\Omega , \mathcal {M})\) be a local minimizer of the functional in (0.1), where \(p(\cdot )\) satisfies assumptions \((\mathrm {P1})\)-\((\mathrm {P2})\), \(k(\cdot )\) satisfies \((\mathrm {K1})\)-\((\mathrm {K2})\) and \(\mathcal {M}\) is as in \((\mathrm {\mathcal {M}1})\)-\((\mathrm {\mathcal {M}2})\). Then there exists a relatively open set \(\Omega _{0}\subset \Omega \) such that \(u \in C^{1,\beta _{0}}_{\mathrm {loc}}(\Omega _{0},\mathcal {M})\) for some \(\beta _{0}\in (0,1)\), and \(\mathcal {H}^{n-\gamma _{1}}(\Omega \setminus \Omega _{0})=0\).

By strengthening further the assumptions on the variable exponent \(p(\cdot )\), we are then able to provide a better dimension estimate for the singular set. This is in the following:

### Theorem 2

- i.
if \(n\le [\gamma _{1}]+1\), then

*u*can have only isolated singularities; - ii.
if \(n>[\gamma _{1}]+1\), then the Hausdorff dimension of the singular set is at the most \(n-[\gamma _{1}]-1\).

*p*-growth we refer to the recent work of Hopper [31]. The extension we make here to the variable exponent case requires a number of non-trivial additional ideas and tools, in particular as far as the dimension estimates stated in Theorem 2 are concerned. This is also related to the recent, aforementioned paper of Tachikawa [46], and it is based on the use of a suitable monotonicity formula. We remark that the variable exponent functional in (0.1) is a significant instance of functionals with (

*p*,

*q*)-growth (following the terminology introduced by Marcellini, [37, 38]). These are variational integrals of the type \(w\mapsto \int F(x,Dw) \ dx\), where the integrand \(F(\cdot )\) satisfies

*x*, since the ellipticity type changes according to the the positivity of the coefficient \(a(\cdot )\). There are several analogies between the variable exponent energy and the double phase one. In particular, one should notice the similarities between the use of the Gehring’s Lemma-based reverse Hölder inequalities made here and the reverse Hölder inequality coming from fractional differentiability exploited in [6, 7]. Moreover, compare the use of localization methods based on

*p*-harmonic type approximation implemented here and in [4]. Such analogies point to a unified approach to non-autonomous functionals with (

*p*,

*q*)-growth conditions, partially implemented in [9]. We plan to investigate this in the context of constrained minimizers in a forthcoming paper [10].

## 2 Notation, main assumptions and functional setting

*c*a general constant larger than one. Different occurrences from line to line will be still indicated by

*c*and relevant dependencies from certain parameters will be emphasized using brackets, i.e.:

*c*(

*n*,

*p*) means that

*c*depends on

*n*and

*p*. We denote \(B_{r}(x_{0}):=\left\{ x \in \mathbb {R}^{n}:|x-x_{0}|<r\right\} \) the open ball centered in \(x_{0}\) with radius \(r>0\); when not relevant, or clear from the context, we shall omit indicating the center: \(B_{r}\equiv B_{r}(x_{0})\). Moreover, for integer \(k\ge 1\), by \(\omega _{k}\) we mean the

*k*-dimensional Lebesgue measure of the unit ball \(B_{1}(0)\subset \mathbb {R}^{k}\). Along the paper,

*k*will assume values

*N*or

*m*. When referring to balls in \(\mathbb {R}^{k}\), \(k\in \{m,N\}\), we will stress it with an apex “

*k*”, i.e.: \(B^ {k}_{r}(a_{0})\) is the open ball with center \(a_{0}\in \mathbb {R}^{k}\) and positive radius

*r*. For \(\alpha ,\beta \in \{1,\cdots ,n\}\) and \(i,j\in \{1,\cdots ,N\}\), we set \(\delta ^{\alpha \beta }\equiv 0\), \(\delta _{ij}\equiv 0\) if \(\alpha \not =\beta \), \(i\not =j\) respectively and \(\delta ^{\alpha \alpha }=\delta _{ii}\equiv 1\). With \(U\subset \mathbb {R}^{n}\) being a measurable subset with positive, finite Lebesgue measure \(0<|U|<\infty \) and with \(f:U\rightarrow \mathbb {R}^{k}\) being a measurable map, we shall denote byits integral average. In particular, when \(U\equiv B_{r}(x_{0})\), we will indicate only the radius and, if necessary, the centre of the ball, i.e.: \((f)_{r}\equiv (f)_{r,x_{0}}:=(f)_{B_{r}(x_{0})}\). For \(g:\Omega \rightarrow \mathbb {R}^{k}\) and \(U\subset \Omega \), with \(\gamma \in (0,1]\) being a given number we shall denote

*g*belongs to the Hölder space \(C^{0,\gamma }(U,\mathbb {R}^{k})\). Let us turn to the main assumptions that will characterize our problem. When considering the functional in (0.1), the exponent \(p(\cdot )\) will always satisfy

*x*] denotes the integer part of

*x*. We refer to Sect. 2 for a detailed description of the geometry of \(\mathcal {M}\). Finally, for shorten the notation we shall collect the main parameters of the problem in the quantity

### Definition 1

It is well known that, under assumptions \((\mathrm {P1})\)-\((\mathrm {P2})\), the set of smooth maps is dense in \(W^{1,p(\cdot )}(\Omega ,\mathbb {R}^{k})\), see e.g. [15, 51, 53, 54]. Following [9, 31] we also recall the analogous definition of such spaces when mappings take values into \(\mathcal {M}\).

### Definition 2

When \(\text{(P1) }\)-\(\text{(P2) }\) and \((\mathrm {\mathcal {M}1})\)-\((\mathrm {\mathcal {M}2})\) are in force, a quick modification of [9, Lemma 6] shows that Lipschitz maps are dense in \(W^{1,p(\cdot )}(\Omega ,\mathcal {M})\) as well. Of course, when \(p(\cdot )\equiv {\text {const}}\), Definitions 1 and 2 reduce to the classical Sobolev spaces \(W^{1,p}(\Omega ,\mathbb {R}^{k})\) and \(W^{1,p}(\Omega ,\mathcal {M})\) respectively. Owing to the \(p(\cdot )\)-growth behavior of the integrand in (0.1), we display our definition of local minimizer.

### Definition 3

In Definition 3, local minimizers are given as maps belonging to the local space \(W^{1,p(\cdot )}_{\mathrm {loc}}(\Omega ,\mathcal {M})\). We stress that, since all the regularity properties of constrained local minimizers treated in this work are of local nature, there is no loss of generality in assuming that \(u \in W^{1,p(\cdot )}(\Omega ,\mathcal {M})\) and that \(x\mapsto k(x)|Du(x)|^{p(x)}\in L^{1}(\Omega )\), see the statements of Theorems 1-2.

### Remark 1

By continuity, all the constants depending on certain fixed values of the map \(p(\cdot )\) are stable when \(p(\cdot )\) varies in the interval \([\gamma _{1},\gamma _{2}]\). Thus, whenever a constant depends on some \(p \in [\gamma _{1},\gamma _{2}]\), this dependence will be denoted by only mentioning \(\gamma _{1}\) and \(\gamma _{2}\), i.e.: \(c(p)\equiv c(\gamma _{1}, \gamma _{2})\).

## 3 Preliminaries

We shall split this section into two parts. In the first one, we collect some basic results concerning the regularity of minimizers of certain type of functionals and in the second one we will give a detailed description of the topology of \(\mathcal {M}\), together with some extension lemmas, which will turn fundamental in order to construct suitable comparison maps in some steps of the proofs of Theorems 1 and 2.

### 3.1 Known regularity results

We start by reporting a Lipschitz estimate for the gradient and a decay estimate for the excess functional of unconstrained local minimizers of functionals of the *p*-laplacean type.

### Proposition 1

The following result is a *p*-harmonic approximation lemma, which will play a crucial role in the proof of Theorem 1. We will state it in the form which better fits our necessities.

### Lemma 1

*p*, such that the following holds. Let \(B_{r}\subset \mathbb {R}^{n}\) be a ball and \(\tilde{B}_{r}\) denote either \(B_{r}\) or \(B_{2r}\). If \(v\in W^{1,p}(\tilde{B}_{r},\mathbb {R}^{k})\) is almost

*p*-harmonic in the sense that

The next are a couple of simple inequalities which will be used several times throughout the paper. They are elementary, see e.g.: [8, 40, 46].

### Lemma 2

- i.
For any \(\varepsilon _{0}>0\), there exists a constant \(c=c(\varepsilon _{0})\) such that for all \(t\ge 0\), \(l\ge m\ge 1\) there holds \( |t^{l}-t^{m}|\le c(l-m)\left( 1+t^{(1+\varepsilon _{0})l}\right) . \)

- ii.
For \(t \in (0,1]\), consider the function \(g(t):=t^{-ct^{\gamma }}\), where \(c>0\) is an absolute constant and \(\gamma \in (0,1]\). Then \(\lim _{t\rightarrow 0}g(t)=1\) and \(\sup _{t \in (0,1]}g(t)\le \exp (c/\gamma )\).

We conclude this section by recalling another fundamental tool in regularity theory, which will help establishing the behavior of certain quantities.

### Lemma 3

[21] Let \(h:[\varrho , R_{0}]\rightarrow \mathbb {R}\) be a non-negative, bounded function and \(0<\theta <1\), \(0\le A\), \(0<\beta \). Assume that \( h(r)\le A(d-r)^{-\beta }+\theta h(d), \) for \(\varrho \le r<d\le R_{0}\). Then \( h(\varrho )\le cA/(R_{0}-\varrho )^{-\beta } \) holds, where \(c=c(\theta , \beta )>0\).

### 3.2 Some extension results

We report some results concerning locally Lipschitz retractions. They have been extensively used in the realm of functionals with *p*-growth, see [27, 28, 31]. For integrands exihibiting (*p*, *q*)-growth they were used for the first time in [9], to prove that if the Lavrentiev phenomenon does not occur in the unconstrained case, then it is absent also in presence of a geometric constraint. According to assumptions \((\mathcal {M}1)\)-\((\mathcal {M}2)\), \(\mathcal {M}\subset \mathbb {R}^{N}\) is a compact, *m*-dimensional \(C^{3}\) Riemannian submanifold, \(\partial \mathcal {M}=\emptyset \) and, in particular, \(\mathcal {M}\) is \([\gamma _{2}]-1\) connected. Let us clarify this concept.

### Definition 4

[31] Given an integer \(j\ge 0\), a manifold \(\mathcal {M}\) is said to be *j*-connected if its first *j* homotopy groups vanish identically, that is \(\pi _{0}(\mathcal {M})=\pi _{1}(\mathcal {M})=\cdots =\pi _{j-1}(\mathcal {M})=\pi _{j}(\mathcal {M})=0\).

It is reasonable to expect some good properties in terms of retractions for this kind of manifolds endowed with a relatively simple topology, as the following lemma shows.

### Lemma 4

Let \(\mathcal {M}\subset \mathbb {R}^{N}\), \(N\ge 3\) be a compact, *j*-connected submanifold for some integer \(j \in \{1,\cdots , N-2\}\) contained in an *N*-dimensional cube *Q*. Then there exists a closed \((N-j-2)\)-dimensional Lipschitz polyhedron \(X\subset Q\setminus \mathcal {M}\) and a locally Lipschitz retraction \(\psi :Q\setminus X \rightarrow \mathcal {M}\) such that for any \(x \in Q\setminus X\), \( |D\psi (x)|\le c/{\text {dist}}(x,X) \) holds, for some positive \(c=c(N,j,\mathcal {M})\).

### Proof

See e.g., [28, Lemma 6.1] for the original proof, or [31, Lemma 4.5] for a simplified version relying on some Lipschitz extension properties of maps between Riemannian manifolds. \(\square \)

A major technical obstruction one can face when dealing with manifold-constrained variational problems is finding comparison maps which satisfy the constraint (notice that, without further regularity details on minimizers, we cannot localize in the image). Precisely, we are no longer allowed to use convex combinations of a minimizer with a suitable cutoff function as to realize valid competitors for the problem. Hence, given any \(w \in W^{1,p(\cdot )}_{\mathrm {loc}}(\Omega , \mathbb {R}^{N})\), we overcome this issue by applying Lemma 4 to assure a local control on the \(L^{p(\cdot )}\)-norm of the gradient of a suitable projected image of *w* in terms of the \(L^{p(\cdot )}\)-norm of *w* itself. This is the content of the next lemma.

### Lemma 5

(Finite energy extension.) Let \(\mathcal {M}\) be as in \((\mathrm {\mathcal {M}1})\)-\((\mathrm {\mathcal {M}2})\) and \(U\Subset \Omega \) an open subset of \(\Omega \) with Lipschitz boundary. Given \(w \in W^{1,p(\cdot )}_{\mathrm {loc}}(\Omega ,\mathbb {R}^{N})\cap L^{\infty }_{\mathrm {loc}}(\Omega , \mathbb {R}^{N})\) with \(w(\partial U)\subset \mathcal {M}\), there exists \(\tilde{w}\in W^{1,p(\cdot )}_{w}(U,\mathcal {M})\) satisfying \( \int _{U}|D\tilde{w}|^{p(x)} \ dx\le c\int _{U}|Dw|^{p(x)} \ dx, \) where \(c=c(N,\mathcal {M},\gamma _{2})\).

### Proof

*x*and, for \(a\in \mathcal {M}\), we denote by \(\text{ reach }(\mathcal {M},a)\) the supremum of the set of all numbers \(r>0\) for which \(\{x\in \mathbb {R}^{N}:|x-a|<r\}\subset \text{ Unp }(A)\). Then, we can set \(\text{ reach }(\mathcal {M}):=\inf _{a\in \mathcal {M}}\text{ Reach }(\mathcal {M},a)\). Notice that, by assumptions \((\mathrm {\mathcal {M}1})\)-\((\mathrm {\mathcal {M}2})\), \(\text{ reach }(\mathcal {M})>0\), see [17, 28, 31]. Now, if for some \(0<\sigma <\mathrm {reach}(\mathcal {M})\), \(V:=\left\{ a \in \mathbb {R}^{N}:{\text {dist}}(a,\mathcal {M})<\sigma \right\} \) is a neighbourhood with the nearest point property, then the metric projection \(\Pi :\overline{V}\rightarrow \mathcal {M}\) associating to any \(a\in \overline{V}\) the unique \(a_{0}\in \mathcal {M}\) such that \({\text {dist}}(a,\mathcal {M})=|a-a_{0}|\), is Lipschitz continuous and \(\overline{V}\) and \(\mathcal {M}\) are homotopy equivalent spaces with \(\pi _{i}(\overline{V})=\pi _{i}(\mathcal {M})\) for all \(i \in \{0,\cdots , [\gamma _{2}]-1\}\), see e.g.: [30, Proposition 1.17] for more details on this matter. Since \(\mathcal {M}\) is compact and

*w*is bounded, there exists an

*N*-dimensional cube

*Q*such that \(\mathcal {M}\subset \overline{V}\subset Q\) and \({\text {dist}}(w,\mathcal {M})\le \frac{1}{2}{\text {dist}}(\mathcal {M},\partial Q)\) almost everywhere. By Lemma 4 with \(j\equiv [\gamma _{2}]-1\), there exists a locally Lipschitz retraction \(\psi :Q\setminus X \rightarrow \overline{V}\) for some \((N-[\gamma _{2}]-1)\)-dimensional Lipschitz polyhedron \(X\subset Q\setminus \overline{V}\), which, by construction stands strictly away from \(\mathcal {M}\). Thus we have a map \(P:=\Pi \circ \psi :Q\setminus X\rightarrow \mathcal {M}\), satisfying

Lemma 5 will be particularly helpful when \(U\) is a ball \(B_{r}\) or an annulus \(B_{r}\setminus B_{\varrho }\) for a proper choice of *r* and \(\varrho \).

## 4 Partial regularity

In this section we first collect a couple of essential inequalities, some basic regularity results stemming only from the minimality condition and then carry out the proof of Theorem 1.

### 4.1 Basic regularity results

The first result is Poincaré’s inequality, well known in the unconstrained case, see [14, Theorem 3.1], and since it is valid for any map \(w \in W^{1,p(\cdot )}_{\mathrm {loc}}(\Omega , \mathbb {R}^{N})\), it transfers verbatim for functions in \(W^{1,p(\cdot )}_{\mathrm {loc}}(\Omega , \mathcal {M})\). However, given that we are dealing with bounded maps (\(\mathcal {M}\) is compact), we present a simplified proof, including also the case in which the domain is an annulus \(A_{r\theta }:=B_{r}\setminus B_{r(1-\theta )}\) for some \(0<\theta <1\).

### Lemma 6

### Proof

*ii.*), (1.1) and the standard Poincaré’s inequality holding for \(p\equiv p_{1}(r)\) we obtain

As to successfully implement Lemma 1, we also need an intrinsic version of Sobolev-Poincaré’s inequality.

### Lemma 7

### Proof

*n*and

*p*, we can conclude that \(c=c(n,N,p)\). Choosing \(d_{1}:=\frac{n\gamma }{n-\gamma p}>1\) since \(\gamma >\frac{n}{n+p}\), and \(d_{2}:=\gamma <1\) we obtain the thesis. Now, if \(p>n\), then there exists \(\gamma \in \left( n/p,1\right) \) so that \(p\gamma >n\). Let \(\kappa :=1-n/(\gamma p)\). From Morrey’s embedding theorem we then havefor \(c=c(n,N,p)\). Fixing \(d_{1}:=\gamma ^{-1}\) and \(d_{2}:=\gamma \) we can conclude.

### Remark 2

Since \(\mathcal {M}\) is compact, for a function *w* taking values in \(\mathcal {M}\) the dependence of the constants appearing in the inequalities in Lemma 6 on the \(L^{\infty }\)-norm of *w* will be expressed as a dependence on \(\mathcal {M}\).

In the following lemma, we present a Caccioppoli-type inequality, which is fundamental for regularity.

### Lemma 8

### Proof

*u*in problem (0.1) and satisfies

*u*, \(\text{(K2) }\), the features of \(\eta \), (3.3) and (1.1) give

*ii*.). \(\square \)

The next step consists in proving an interior higher integrability result for local minimizers of (0.1).

### Lemma 9

### Proof

*ii.*) and Hölder’s inequality. Now, an application of Gehring-Giaquinta-Modica’s lemma, [26, Chapter 6] renders the existence of a positive \(\tilde{\delta }_{0}=\tilde{\delta }_{0}(n,N,\mathcal {M},\lambda ,\Lambda ,\gamma _{1}, \gamma _{2},[p]_{0,\alpha },\alpha )\) so thatwith \(c=c(n,N,\mathcal {M},\lambda ,\Lambda ,\gamma _{1},\gamma _{2},[p]_{0,\alpha },\alpha )\), for all \(\delta \in [0,\tilde{\delta }_{0})\). Finally, after a standard covering argument, we obtain that \(|Du|^{(1+\delta )p(\cdot )}\in L^{1}_{\mathrm {loc}}(\Omega )\) for all \(\delta \in [0,\tilde{\delta }_{0})\). \(\square \)

### Remark 3

The following lemma is an up to the boundary higher integrability result. The argument is well-known to specialists, see [1, 40], and it essentially relies on the fact that Caccioppoli’s inequality can be carried up to the boundary. However, we did not manage to find in the literature a proof for the manifold-constrained case, so we shall report it here.

### Lemma 10

### Proof

*v*on \(\partial B_{r}\) and on \(\partial (B_{r}\cap B_{t}(x_{0}))\) in the sense of traces, so Lemma 5 with \(p(\cdot )\) equal to constant

*p*, provides us with a map \(\tilde{w}\in W^{1,p}_{v}(B_{r}\cap B_{t}(x_{0}),\mathcal {M})\) such that

*v*in the Dirichlet class \(W^{1,p}_{v}(B_{r}\cap B_{t}(x_{0}),\mathcal {M})\) and (3.5) render

*p*),for \(c=c(n,N,\mathcal {M},\lambda ,\Lambda ,\gamma _{1},\gamma _{2})\). The two cases can be combined via a standard covering argument. Precisely, upon defining

*v*within the Dirichlet class \(W^{1,p}_{u}(B_{r},\mathcal {M})\) we can conclude thatwith \(c=c(n,N,\mathcal {M},\lambda ,\Lambda ,\gamma _{1},\gamma _{2})\). \(\square \)

The next corollary allows recovering some useful estimates for the average of the gradient of solutions to problem (0.1).

### Corollary 3

### Proof

*u*. In fact we havefor \(c=c(n,N,\mathcal {M},\lambda ,\Lambda ,\gamma _{1},\gamma _{2},[p]_{0,\alpha },\alpha )\). On the other hand, combining Lemmas 9 and 8, we havewith \(c=c(n,N,\mathcal {M},\lambda ,\Lambda ,\gamma _{1},\gamma _{2},[p]_{0,\alpha },\alpha )\), for any \(\delta \in (0,\tilde{\delta }_{0})\).

### 4.2 Proof of Theorem 1

Now we are ready to prove Theorem 1. For the reader’s convenience, we shall split the proof in seven steps.

*Step 1: comparison, first time.*We define \(\delta _{0}:=\frac{1}{2}\min \left\{ \tilde{\delta }_{0},1\right\} \), where \(\tilde{\delta }_{0}\) is the higher integrability threshold from Lemma 9. Notice that by \(\text{(P1) }\), the set

*Step 7*, so, from now on, \(\gamma _{2}<n\) holds. We set

*v*solves the Euler-Lagrange equation

*P*2)-(

*K*2), (3.19)\(_{2}\) and Hölder’s inequality we estimate

*ii.*) we havewith \(c=c(n,N,\mathcal {M},\lambda ,\Lambda ,\gamma _{1},\gamma _{2},[p]_{0,\alpha },\alpha )\). On the other hand, by Poincaré’s inequality, the minimality of

*v*in class \(W^{1,p_{2}(2r)}_{u}(B_{r},\mathcal {M})\) and (3.16) we boundfor \(c=c(n,N,\lambda ,\Lambda ,\gamma _{1},\gamma _{2})\). Inserting the content of the previous two displays in (3.21) we obtain

*u*, we see that

*i.*) with \(\varepsilon _{0}\equiv \sigma _{0}/2\), (3.14), Lemma 9 and (3.16) we obtainwhere \(c=c(\texttt {data})\) and \(\kappa :=\alpha -\gamma _{2}\sigma _{0}>0\) because of (3.12). Choosing now \(\sigma \) as in (3.20), using also Lemma 10 and (3.14), in a totally similar way we getwhere we set \(\varepsilon _{0}\equiv \sigma \) while applying Lemma 2 (

*i.*). Here \(c=c(\texttt {data})\) and \(\kappa >0\) is as before. All in all, remembering that, by definition, \(0<\sigma <\sigma _{0}\), we can conclude

*v*we obtain

*Step 2: harmonic approximation.*We aim to apply Lemma 1 in order to obtain an unconstrained \(p_{2}(2r)\)-harmonic map suitably close to

*v*. Hence, we need to transfer condition (3.16) from

*u*to

*v*. From the minimality of

*v*in class \(W^{1,p_{2}(2r)}_{u}(B_{r},\mathcal {M})\) and (3.16) we see that

*v*is approximately \(p_{2}(2r)\)-harmonic in the sense of (2.3). This is actually the case: in fact, with reference to the terminology used in Lemma 1, let \(\tilde{d}\equiv d_{2}\), where \(d_{2}\in (0,1)\) is the exponent given by Lemma 7, pick any \(\tilde{\theta }\in (0,1)\) and let \(\tilde{\delta }=\tilde{\delta }(\tilde{\theta },\tilde{d},p_{2}(2r))\) be the “closeness” parameter appearing in (2.3). Moreover, for reasons that will be clear in a few lines, we also impose a first restriction on the size of \(\varepsilon \). Precisely, keeping in mind the definition of \(c_{*}\), we ask that

*Step 3: comparison, second time*. First notice that, since

*v*is a solution to the frozen Dirichlet problem (3.17), a Caccioppoli-type inequality holds. In fact, with the same strategy adopted for the proof of Lemma 8, we have

*v*in the Dirichlet class \(W^{1,p_{2}(2r)}_{u}(B_{r},\mathcal {M})\) and of \(\tilde{h}\) in \(v+W^{1,p_{2}(2r)}_{0}(B_{r},\mathbb {R}^{N})\) and the reference estimate in (2.2)\(_{1}\) which holds for \(\tilde{h}\) since \(\tilde{h}\) is a solution to (2.4) with \(p\equiv p_{2}(2r)\), (set \(k_{0}\equiv 1\), \(p\equiv p_{2}(2r)\), \(g^{\alpha \beta }\equiv \delta ^{\alpha \beta }\) and \(h_{ij}\equiv \delta _{ij}\) for \(\alpha ,\beta \in \{1,\cdots ,n\}\) and \(i,j\in \{1,\cdots ,N\}\) in (2.1)). Here \(c=c(\texttt {data})\). For the ease of exposition, let us set \(s\equiv 2r\). Hence we can rewrite the previous estimate as

*Step 4: Morrey-type estimates.*Our goal now is to prove a Morrey type estimate for the energy which will eventually lead to the continuity of solutions. For \(\tau \in \left( 0,\frac{1}{4}\right) \), we let \(\varrho \equiv \tau s\) in (3.32) and multiply both sides of it by \((\tau s)^{p_{2}(s)-n}\). We then have

*s*, \(\varepsilon \), \(\tilde{\theta }\in (0,1)\), from the definitions of \(\kappa _{1}\), \(\kappa _{2}\) and \(\kappa _{3}\) we have

*Step 1*satisfies \(c_{3}(\tau R_{*})^{\beta }\le (\varepsilon /5)\). Finally we recall that \(\tilde{\theta }\) is arbitrary, therefore we fix \(\tilde{\theta }=2^{-\gamma _{2}}\tau ^{\frac{n\gamma _{2}}{d_{2}\gamma _{1}}}\) and, since \(\tilde{\kappa }_{2}\ge \tilde{\kappa }_{3}\) renders \(\varepsilon ^{\tilde{\kappa }_{2}}\le \varepsilon ^{\tilde{\kappa }_{3}}\), the choice

*s*by \(\tau s\), \(\tau ^{2}s\), \(\tau ^{3}s\), \(\cdots \) to get

*y*belonging to a sufficiently small neighborhood of \(x_{0}\). Hence, if we let

*Step 5: Hausdorff dimension of the Singular Set.*Given the characterization of \(D_{0}\), we easily see that, if \(\Sigma _{0}(u,B_{\tilde{R}_{*}}(x_{0})):=B_{\tilde{R}_{*}}(x_{0})\setminus D_{0}\), then

*Step 6: partial*\(C^{1,\beta _{0}}\)-regularity. In this part we follow the approach of [28, Theorem 3.1]. So far we know that the regular set \(\Omega _{0}\subset \Omega \) is a relatively open set of full

*n*-dimensional Lebesgue measure and \(u\in C^{0,\beta }_{\mathrm {loc}}(\Omega _{0},\mathcal {M})\) for all \(\beta \in (0,1)\). For reasons that will be clear in a few lines, we fix

*u*and up to scaling, rotating and translating \(\mathcal {M}\) we can now assume that \(u(\tilde{\Omega })\) is contained into the image of a single chart \(f(B_{1}^{m})\), so we can find an \(\omega :\tilde{\Omega }\rightarrow \mathbb {R}^{m}\) such that \(u=f(\omega )\) and \(|\omega |\le 1\). Here \(f:\mathbb {R}^{m}\mapsto \mathcal {M}\) is such that

*u*is an \(\mathcal {M}\)-constrained local minimizer of (0.1), then \(\omega \) minimizes the variational integral

*f*a chart, \((h_{ij})_{ij}\) is uniformly elliptic and uniformly bounded, in the sense that

*c*, \(c_{1}\), \(c_{2}\) depend only on \(\mathcal {M}\). Given the previous considerations, it is easy to see that the integrand

*m*, \(\mathcal {M}\), \(\lambda \), \(\Lambda \), \(\gamma _{1}\), \(\gamma _{2}\), \([k]_{0,\alpha }\), \([p]_{0,\alpha }\) and \(\alpha \), except for \(c_{\varepsilon _{0}}\), which, in addition, depends also from \(\varepsilon _{0}\). In particular, from (3.51)\(_{1}\), we see that \(\omega \) minimizes a functional controlled from below and above by the \(p(\cdot )\)-laplacean energy, so there is no loss of generality in assuming that Lemmas 8 and 9 (and 10 for the associated frozen problem) hold true with the same parameters as before. Moreover, (3.48)\(_{3}\), (3.49) and (3.45) allow transferring regularity from

*u*to \(\omega \). In fact we have

*p*(

*x*)) and (3.54) we then estimate

*ii.*), 8 and by the minimality of \(\vartheta \) we see that

*Step 1*, estimates (3.25)-(3.26) we can conclude that

*Step 7: the case*\(p(\cdot )>n-\delta _{0}/2\). As mentioned in

*Step 1*, \(u\in C^{0,\beta '}(\Omega ^{+},\mathcal {M})\), with \(\beta ':=\frac{\delta _{0}}{4n+\delta _{0}}\), so we no longer need to impose a smallness condition like (3.16). Being \(p(\cdot )\) continuous, \(\Omega ^{+}\) is open, so we can fix a ball \(B_{\tilde{R}_{*}}\equiv B_{\tilde{R}_{*}}(x_{0})\Subset \Omega ^{+}\) with \(\tilde{R}_{*}\) satisfying (3.13). Let \(\sigma _{0}\) be as in (3.12), so (3.14) is matched on all balls \(B_{4\varrho }\subset B_{R_{*}}\subset B_{\tilde{R}_{*}}\), where the size of \(R_{*}\le \tilde{R}_{*}/2\) will be specified later on. As we did in

*Step 6*, we fix open subsets \(\tilde{\Omega }\Subset \Omega '\Subset \Omega ^{+}\) and cover \(\tilde{\Omega }\) with a finite number of balls contained inside \(\Omega '\) whose size and number will now depend on \(\mathcal {M}\), on \([u]_{0,\beta ';\tilde{\Omega }}\) and on \({{\,\mathrm{diam}\,}}(\Omega ')\), having images contained in small coordinate neighborhoods of \(\mathcal {M}\). Again we can find \(\omega \in W^{1,p(\cdot )}(\tilde{\Omega },\mathbb {R}^{m})\cap C^{0,\beta '}(\tilde{\Omega },\mathbb {R}^{m})\), unconstrained local minimizer of the variational integral (3.50) with integrand \(H(\cdot )\) matching (3.51), such that \(|\omega |\le 1\), \(u=f(\omega )\) where

*f*is as in (3.48). Our goal is to show the validity of a Morrey decay estimate like (3.52)\(_{3}\). To do so, fix \(B_{4\varrho }\Subset B_{R_{*}}\) and let \(\vartheta \in W^{1,p_{2}(\varrho )}(B_{\varrho /4},\mathbb {R}^{m})\) be a solution to the frozen Dirichlet problem (3.53). Notice that the estimates obtained in

*Step 6*till (3.64) do not require any specific value of \(\beta \), therefore, by (3.64) with \(\tilde{\beta }\) replaced by \(\beta '\) we immediately have

*Step 4*, clearly with \(\omega \) instead of

*u*, we readily have

*j*there holds

*Step 4*, (3.41) we can extend (3.72) to the full range \(0<\varsigma <\varrho \) and, proceeding as in estimates (3.42)-(3.43) we can get rid of the restriction \(s\le R_{*}\); as already mentioned, we shall only retain \(s\le \tilde{R}_{*}/2\). Furthermore, it directly implies thatfor \(c=c(\texttt {data},||(|Du|)^{p(\cdot )} ||_{L^{1}(\tilde{\Omega })}, [u]_{0,\beta ';\tilde{\Omega }},\beta )\) and therefore, being \(\beta \in (0,1)\) arbitrary, by Morrey’s growth theorem and a standard covering argument, we can conclude that \(\omega \in C^{0,\beta }_{\mathrm {loc}}(\Omega ^{+},\mathbb {R}^{m})\) for any \(\beta \in (0,1)\). Now, for all \(B_{4\varsigma }\Subset \Omega ^{+}\) such that \(0<\varsigma \le \tilde{R}_{*}/2\), by Lemmas 9, 8 and 2 (

*ii*.), we obtain

*Step 6*.

## 5 Dimension reduction

In this section we obtain a further reduction of the dimension of the singular set of *p*(*x*)-harmonic maps, for \(p(\cdot )\ge 2\) Lipschitz continuous, thus improving, at least in this case, the result given in Theorem 1, *Step 5*.

### 5.1 Compactness of minimizers and Monotonicity formula

The proof of Theorem 2 essentially needs two components to be carried out. The first is the compactness of sequences of minimizers of (0.1) under uniform assumptions, while the second is the monotonicity along solutions to (0.1) of a certain quantity strictly related to the *p*(*x*)-energy. Those arguments are quite classical, see e. g. [23, 27, 46].

### Lemma 11

*v*is a constrained local minimizer of the functional

*v*.

### Proof

The proof is divided into three steps.

*Step 1: weak*\(W^{(1+\tilde{\sigma })p_{0}}\)

*-convergence.*Since \(\mathcal {M}\) is compact, \(\sup _{j\in \mathbb {N}}||u_{j} ||_{L^{\infty }(B_{1})}\le c(\mathcal {M})\), and given that \(\gamma _{1}>1\), we obtain, up to (non relabelled) subsequences,

*j*. By Lemma 9, we know that \((u_{j})_{j \in \mathbb {N}}\subset W^{1,(1+\delta )p(\cdot )}(B_{1},\mathcal {M})\) for all \(\delta \in (0,\tilde{\delta }_{0})\). Let \(\delta _{2}:=\frac{1}{4}\min \{\tilde{\sigma }_{0},\tilde{\delta }_{0}\}\), where \(\tilde{\sigma }_{0}\) is the higher integrability threshold given by Lemma 10 and pick any \(\delta \in (0,\delta _{2})\). Because of the uniform convergence of the \(p_{j}\)’s to the constant \(p_{0}\), taking

*j*sufficiently large we can find positive constants \(\gamma _{1}\le q_{1}\le q_{2}\le \gamma _{2}\) such that

*Step 2: compactness.*We aim to show that

*v*is an \(\mathcal {M}\)-constrained local minimizer of \(\mathcal {E}_{0}\). To do so, we first claim that

*i.*) with \(\varepsilon _{0}\equiv \delta /2\) and (4.1) we have

*v*outside \(B_{r}\). In this way, \(\tilde{v}\in W^{1,p_{0}}_{\mathrm {loc}}(B_{1},\mathcal {M})\cap W^{1,p_{0}}_{v}(B_{r},\mathcal {M})\). Since we are assuming that \((p_{j})_{j\in \mathbb {N}}\) converges uniformly to \(p_{0}\) on \(B_{1}\), we can take \(j \in \mathbb {N}\) so large that

*i.*) with \(\varepsilon _{0}\equiv \delta /4\) we see that

*j*large enough such that

*v*is an \(\mathcal {M}\)-constrained local minimizer of \(\mathcal {E}_{0}\) and, as a direct consequence of the last chain of inequalities, \(\mathcal {E}_{j}(u_{j},B_{r})\rightarrow \mathcal {E}_{0}(v,B_{r})\).

*Step 3: singular points.* Once we have the results contained in *Steps 1*-*2* by hand, the proof of *Step 3* goes as the one in [46, Lemma 3.1] and we shall omit it. \(\square \)

We stress that Lemma 11 holds with \(p(\cdot )\ge \gamma _{1}>1\) Hölder continuous rather than Lipschitz. We need stronger assumptions only to prove a suitable monotonicity formula.

### Lemma 12

### Proof

*v*introduced during the proof of Lemma 4.1 to obtain estimate (4.17) must be replaced by a solution to the Dirichlet problem

### 5.2 Proof of Theorem 2

Combining the compactness Lemma 11 and the monotonicity formula obtained in Lemma 12, we are ready to prove Theorem 2. If \(\Omega ^{+}\) is as in (3.11), then \(u \in W^{1,n+\delta _{0}/4}(\Omega ^{+},\mathcal {M})\). So, by Morrey’s embedding theorem, \(u \in C^{0,\beta '}(\Omega ^{+},\mathcal {M})\) for \(\beta ':=\delta _{0}/(4n+\delta _{0})\) and, by *Step 7* of Theorem 1, we can conclude that *Du* is locally \(\beta _{0}\)-Hölder continuous on \(\Omega ^{+}\) for some \(\beta _{0}\in (0,1)\). This observation shows that, to prove Theorem 2 it is enough to assume that \(\gamma _{2}<n\), and this condition assures the applicability of Lemma 12.

*Case 1:*\(n\le [\gamma _{1}]+1\). Since \(t\mapsto \Phi (t)\) can be seen as a difference between an increasing function of

*t*and \(c t^{\gamma }\) for some \(\gamma \in (0,1)\) and a positive constant

*c*, it admits a finite limit as \(t\rightarrow 0\). Assume that

*u*has a singular point at \(\bar{x}=0\) which is not isolated. Then we can find a sequence of singular points \((x_{j})_{j\in \mathbb {N}}\) such that \(x_{j}\rightarrow 0\). Setting \(R_{j}:=2|x_{j}|<T<1\) we see that, for any

*j*the scaled function \(u_{j}(x):=u(R_{j}x)\) is a constrained local minimizer of the functional

*v*, constrained local minimizer of \(\mathcal {E}_{0}(w,B_{1}):=\int _{B_{1}}|Dw|^{p(0)} \ dx\) and that the \(y_{j}\)’s converge to \(\bar{y}\), singular point of

*v*with \(|\bar{y}|=1/2\). Now pick two constants \(0<\lambda<\mu <1\) and apply Lemma 12 with \(r\equiv \lambda R_{j}\) and \(R\equiv \mu R_{j}\) to get

*v*is homogeneous of degree 0, so the whole segment joining \(\bar{x}\) and \(\bar{y}\) is made of singular points of

*v*, but, since we are assuming \(n\le [\gamma _{1}]+1\le [p(0)]+1\), we obtain a contradiction to [28, Theorem 4.5], which states that, under these conditions,

*v*can have only isolated singularities.

*Case 2:* \(n>[\gamma _{1}]+1\). Let us assume that for some \(l>0\), \(\mathcal {H}^{l}(\Sigma _{0}(u))>0\). Then, by blowing up, we obtain a constrained local minimizer *v* of \(\mathcal {E}_{0}\) with \(\mathcal {H}^{l}(\Sigma _{0}(v))>0\), (see [23], Chapter 10). On the other hand, by [28, Theorem 4.5], \(l<n-[p(0)]-1\le n-[\gamma _{1}]-1\) and this concludes the proof.

## Notes

### Acknowledgements

The author would like to thank the referees for the patient they had in reading the originally submitted manuscript and for the their most useful remarks, that eventually led to an improved version of the paper. This work was supported by the *Engineering and Physical Sciences Research Council* [*EP* / *L*015811 / 1].

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