Boundary regularity for manifold constrained $p(x)$-harmonic maps

We prove partial and full boundary regularity for manifold constrained $p(x)$-harmonic maps.


Introduction
In this paper we complete the partial regularity theory for p(x)-harmonic maps studied in [9] providing partial and full boundary regularity for manifold constrained minima of the variable exponent energy: g + W 1,p(·) (Ω, M) ∩ W 1,p(·) 0 (Ω, R N ) ∋ w → E(w, Ω) := Ω k(x)|Dw| p(x) dx (1.1) for a suitable boundary datum g :Ω → M. Our main accomplishment is that there exists a relatively (toΩ) open subset Ω 0 ⊂Ω of full n-dimensional Lebesgue measure on which u is locally Hölder continuous and the singular set Σ 0 :=Ω \ Ω 0 has Hausdorff dimension at the most equal to n − γ 1 , see (2.2) 1 below for more informations on this quantity. This is the content of the following theorem. Moreover, after strengthening the hypotheses on the variable exponent p(·) and on the boundary datum g(·), we can prove that the singular set of solutions to problem g + W 1,p(·) (Ω, M) ∩ W 1,p(·) 0 (Ω, R N ) ∋ w → J(w, Ω) := Ω |Dw| p(x) dx (1. 2) does not intersect the boundary ∂Ω. In this respect we have Theorem 2. Under assumptions (2.1), (2.4) and (2.6), let u ∈ W 1,p(·) (Ω, M) be a solution to the Dirichlet problem (1.2) with boundary datum g :Ω → M satisfying (2.7). Then there exists a constant Υ ≡ Υ(data) ∈ (0, 1] such that if [g] 0,1− n q ;Ω < Υ, (1.3) then Σ 0 ⋐ Ω and so u is 1 − n q -Hölder continuous in a neighborhood of ∂Ω. We immediately refer to Section 2.2 for the complete list of assumptions in force concerning the regularity of ∂Ω, the coefficients appearing in the energies displayed in (1.1)-(1.2) and the topology of the manifold M. The results exposed in Theorems 1-2 are new already in the case p(·) ≡ const. In fact, we recover for the p(x)-Laplacian the boundary regularity theory already available for p-harmonic maps, under weaker assumptions on the boundary datum than those considered in [22,30,50]. Let us put our results into the context of the available literature. The regularity theory for vector-valued minimizers of functionals modelled upon the p-Laplacean integral, i.e. variational problems like started with the seminal paper [53] and received several contributions later on, see [23-25, 28, 39, 42] and references therein for an overview of the state of the art concerning p-laplacean type problems. On the other hand, the regularity theory in the case when both minimizers and competitors take values into a manifold M ⊂ R N faces additional difficulties. The cornerstones of the theory were laid down by the fundamental papers [16,18,49,50] analyzing harmonic maps, i.e., constrained minimizers of the functional in (1.4) for p = 2, see also [51]. We mention also the recent works [45,46] for a fine analysis of the singular set of harmonic maps. The extension of such basic results to the case p = 2 has been done in the by now classical papers: [20-22, 30, 41]. Moreover, several results have been extended to more general functionals with p-growth, for instance the quasiconvex case has been treated in [36] while a purely PDE approach has been proposed in [15]. The matter of boundary regularity for vectorial problems is rather delicate and received lots of attention in the literature, starting from [37], which covers the case of quadratic functionals. This theory has been extended later on to variational integrals of p-laplacean type, see [13] for the first results in this direction and [3,14,26,28,38] for general systems with standard p-growth. On the other hand, we notice that energies of the type in (1.1) do not satisfy conditions as in (1.4), but rather, the more general and flexible one The systematic study of functionals as in (1.5) started in [43,44] and, subsequently, has undergone an intensive development over the last years, see for instance [2, 4-6, 11, 17, 19, 32, 33, 35]. In particular, the energy in (1.1) have been introduced in the setting of Calculus of Variations and Homogenization in the seminal works [54][55][56]. Energies as in (1.1) also occur in the modelling of electro-rheological fluids, a class of non-newtonian fluids whose viscosity properties are influenced by the presence of external electromagnetic fields [1], see also [12] for the basic properties of the p(x)-Laplacian. As for regularity, the first result in the vectorial case has been obtained in [8], where it is shown that local minimizers of energy (1.2) are locally C 1,β -regular in the unconstrained case. Subsequently, the regularity theory of functionals with variable growth has been developed in a series of interesting papers, [47,48,52], where the authors established partial regularity results for unconstrained minimizers that are on the other hand obviously related to the constrained case. Especially, in [52] is given 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. Finally, [9] is devoted to the study of partial inner regularity of manifold constrained p(x)-harmonic maps and to the analysis and dimension-reduction of their singular set.
Organization of the paper. This paper is organized as follows: Section 2 contains our notation, the list of the assumptions which will rule problems (1.1)-(1.2), several by now classical tools in the framework of regularity theory and some results of geometric and topological nature on Lipschitz retractions. Finally, Sections 3-4 are devoted to the proof of Theorem 1 and Theorem 2 respectively.

Preliminaries
In this section we display our notation, list the main assumptions in force throughout the paper and collect some useful tools for regularity theory and several well-known results in the framework of manifold-valued maps.
2.1. Notation. Following a usual custom, we denote by c a general constant larger than one. Different occurrences from line to line will be still denoted by c, while special occurrences will be denoted by c 1 , c 2 ,c or the like. Relevant dependencies on parameters will be emphasized using parentheses, i.e., c ≡ c(p, ν, L) means that c depends on p, ν, L. Given any measurable subset U ⊂ R n , we denote by |U | its n-dimensional Lebesgue measure and with H k (U ) its kdimensional Hausdorff measure, for some k ≥ 0. For a point x 0 ∈ R n and a number ̺ > 0 we indicate with B ̺ (x 0 ) := x ∈ R n : |x − x 0 | < ̺ the open ball centered at x 0 and with radius ̺ and further, B ̺ ≡ B ̺ (0). Similarly, for x 0 ∈ R n−1 × {0} we define the half ball centered at x 0 as: . We also name Γ ̺ (x 0 ) the set x ∈ R n : x n = 0 and |x 0 − x| < ̺ and ∂ + B + ̺ (x 0 ) := ∂B + ̺ (x 0 ) \ Γ ̺ (x 0 ). As before, Γ ̺ ≡ Γ ̺ (0). With U ⊂ R n being a measurable subset having finite and positive n-dimensional Lebesgue measure, and with h : U → R k , being a measurable map, we shall denote by its integral average. Similarly, with γ ∈ (0, 1) we denote the Hölder seminorm of h as It is well known that the quantity defined above is a seminorm and when [h] 0,γ;U < ∞, we will say that h belongs to the Hölder space C 0,γ (U, R k ). When clear from the context, we will omit the reference to U , i. When considering the functional in (1.1), the exponent p(·) will always satisfy p ∈ C 0,α (Ω) for some α ∈ (0, 1] 1 < γ 1 := inf x∈Ω p(x) ≤ p(x) ≤ γ 2 := sup x∈Ω p(x) < ∞, holds true. We anticipate that in the estimates contained in Section 3.2, only min {α, ν} will be relevant, so, for simplicity, for the proof of Theorem 1 we will assume that α = ν, i.e.: p(·), k(·) ∈ C 0,α (Ω). When dealing with the question of full boundary regularity, we need higher regularity for p(·). Precisely, we shall suppose that with γ 1 and γ 2 as in (2.2) 2 . Given an half ball B + R and a ball B ̺ (x 0 ) with x 0 ∈ B + R and ̺ ∈ (0, R − |x 0 |), we denote Since in (2.5) we will always consider the intersection with the same ball B + R , the reference to R in the symbols p 1 , p 2 is omitted. When clear from the context, in (2.5) we shall not mention With a little abuse, we will adopt the notation in (2.5) also to denote the infimum (resp. the supremum) of p(·) on B + R : the context will remove any ambiguity. Notice that there is no loss of generality in assuming γ 1 < γ 2 , otherwise p(·) ≡ const onΩ, and in this case the problem is very well understood, [22,30,50]. Furthermore, we need to impose some topological restriction on the manifold M. Precisely, we ask that Here [x] denotes the integer part of x and the definition of j-connectedness is given in Section 2.4, Definition 4. Moreover, we assume that the boundary datum satisfies: Combining (2.7) with Morrey embedding theorem we automatically get that Finally, to shorten the notation we shall collect the main parameters of the problem in the quantities data p(·) := (n, N, M, λ, Λ, γ 1 , γ 2 , q, [p] 0,α , α); Any dependencies of the constants appearing in the forthcoming estimates from quantities depending on the characteristics of M, such as, for instance, the L ∞ -norm of maps with range in M (which is clearly finite being M compact) will be simply denoted as a dependency from M in the form: c ≡ c(M).
Remark 2.1. Assumption (2.1) assures that there exists a positive constantr ≡r(n, Ω) such that B ̺ (x 0 ) ∩ Ω is simply connected for all ̺ ∈ (0,r] and any x 0 ∈ ∂Ω. This renders the existence of a positive constant c ≡ c(n, Ω) such that with constants implicit in "∼" depending on n, Ω. We shall refer to such constants with the term "Ahlfors constants", see [13,Section 2].
As to fully clarify the framework we are going to adopt, we need to introduce some basic terminology on the so-called Musielak-Orlicz-Sobolev spaces. Essentially, these are Sobolev spaces defined by the fact that the distributional derivatives lie in a suitable Musielak-Orlicz space, rather than in a Lebesgue space as usual. Classical Sobolev spaces are then a particular case. Such spaces and related variational problems are discussed for instance in [7,12,31,57], to which we refer for more details. Here, we will consider spaces related to the variable exponent case in both unconstrained and manifold-constrained settings.
It is well known that, under assumptions (2.2), the set of smooth maps is dense in W 1,p(·) (Ω, R k ), see e.g. [17,57]. Following [9] we also recall the analogous definition of such spaces when mappings take values into M.  Of course, when p(·) ≡ const, Definitions 1 and 2 reduce to the classical Sobolev spaces W 1,p (Ω, R k ) and W 1,p (Ω, M) respectively. Since the regularity question in Ω is local in nature, we can choose coordinates {x i } n i=1 centered at x 0 ∈ ∂Ω such that locally Ω is the upper half space R n ∩ {x n > 0}, therefore, to avoid unnecessary complications, from now on we will assume that Ω ≡ B + 1 , see [13,14,30,37,38,50] for a more detailed discussion on this matter. Let us display the definition of constrained W 1,p(·) -minimizer of (1.1) in B + 1 . and for all maps w ∈ W 1,p(·) (B + 1 , M) so that (u − w) ∈ W 1,p(·) 0 (B + 1 , R N ). The conditions displayed above define class C p(·) g (B + 1 , M).

2.3.
Well-known results. When dealing with p-Laplacean type problems, we shall often use the auxiliary vector fields V s,t : R N ×n → R N ×n , defined by V s,t (z) := (s 2 + |z| 2 ) (t−2)/4 z, t ∈ (1, ∞) and s ∈ [0, 1] (2.9) whenever z ∈ R N ×n . If s = 0 we shall simply write V s,t ≡ V t . A useful related inequality is contained in the following where the equivalence holds up to constants depending only on n, k, t. An important property which is usually related to such field is recorded in the following lemma.
The next are a couple of simple inequalities which will be used several times throughout the paper. They are elementary, see e.g.: [8,9,47,52].
We conclude this section by recalling the celebrated iteration lemma, [25].

Extensions.
In this section we shall borrow from [9] some useful lemmas concerning locally Lipschitz retractions. Such results were first introduced in [30] and intensively used in the literature for dealing with possibly non-homogeneous variational problems whose structure is a priori non-compatible with any kind of monotonicity formulae, [10,36]. We refer to Remark 2.2 below for a quick discussion on this matter. We start with clarifying a key assumption in our paper, which is the concept of j-connectedness.
It is well-known that a compact manifold M ⊂ R N without boundary admits a tubular neighborhood M ⊂ ω ⊂ R N . Identifying M with its image in R N , we say that a neighborhood ω of M has the nearest point property if for every x ∈ ω there is a unique point Π M (x) ∈ M such that dist(x, M) = |x − Π M (x)|. The map Π M : ω → M is called the retraction onto M, we shall refer to it also as "projector". Moreover, the regularity of M influences the regularity of Π M in the following way: see [36] for a deeper discussion on this matter. It is important to stress that manifolds endowed with the relatively simple topology described by Definition 4 enjoy good properties in terms of retractions.
Lemma 2.4. Let M ⊂ R N be a compact, j-connected submanifold for some integer j ∈ {0, · · · , N − 2} contained in an N -dimensional cube Q. Then there exists a closed (N − j − 2)-dimensional Lipschitz polyhedron X ⊂ Q \ M and a locally Lipschitz retraction ψ : Q \ X → M such that for any x ∈ Q \ X, |Dψ(x)| ≤ c/ dist(x, X) holds, for some positive c ≡ c(N, j, M).
Proof. We refer to [30, Lemma 6.1] for the original proof, or [36,Lemma 4.5] for a simplified version relying on some Lipschitz extensions of maps between Riemannian manifolds.
The next lemma allows modifying the image of a map while keeping under control boundary values and p(·)-energy, see also [9,Lemma 5].
Lemma 2.5. Let M be as in (2.6) and U ⊆ B + 1 a subset with positive measure and piecewise Remark 2.2. When dealing with manifold constrained minima of the p-Laplacean energy it is customary to recover the fundamental Caccioppoli inequality by exploiting the so-called monotonicity formula, see [20-22, 41, 49-51]. This way cannot be used in our case. Even though it is possible to show a monotonicity formula for the p(x)-energy, Lemma 4.2 below, see also [52,Lemma 4.1] or [9,Lemma 12], its proof crucially requires some corollaries of Gehring Lemma, which, in turn, is implied by Caccioppoli inequality, whose proof requires the monotonicity formula. Lemma 2.5 breaks this vicious circle giving the chance of deriving Caccioppoli inequality directly by minimality, as we will see in Section 3.1.

Partial boundary regularity
As mentioned in Section 2.2, to avoid unnecessary complications, we shall take Ω ≡ B + 1 . In fact, since ∂Ω is C 2 -regular, given any x 0 ∈ ∂Ω, there exists an open neighborhood B x0 of x 0 and a change of variable Ψ 0 ∈ C 2 (B x0 , R n ) so that in the new coordinates y i := Ψ i 0 (x) there holds that Moreover, there exists a positive constant c 0 ≡ c 0 (n, ∂Ω) such that We stress that, being ∂Ω compact, the constant c 0 does not depend from x 0 . A straightforward computation shows that, if u ∈ W 1,p(·) (Ω, M) solves (1.1), then the mapũ := u • Ψ −1 0 solves an analogous problem still satisfying (2.2) and (2.3). Assumption (2.7) on the boundary condition is preserved as well: . We refer to [13,30,37] for more details on this matter. Therefore, keeping Definition 3 in mind, we shall study problem with k(·) and p(·) as in (2.3)-(2.2) respectively and g as in (2.7).
3.1. Basic regularity results. We first fix a threshold radius R * ∈ (0, 1] so that and choose a R ∈ (0, R * ). Further restrictions on the size of R * will be imposed in Section 3.2. An immediate consequence of (3.2) is that, given any half-ball B + R and all balls B ̺ (x 0 ) with which is, on the other hand, automatic when p 1 (x 0 , ̺) ≥ n. Obviously, in (3.3) we adopted the usual terminology for c ≡ c(n, N, p,ĉ). Here p * := max 1, np n+p . We consider now an intrinsic version of [13, Theorem 2.4].
Proposition 3.1. Let U ⊂ R n be an open, bounded domain with piecewise C 1 -regular boundary and finite Ahlfors constants depending only from n. Let also A ⊂Ū be a closed subset. Consider two non-negative functions f 1 ∈ L 1 (U ) and f 2 ∈ L 1+σ (U ) for someσ > 0. With θ ∈ (0, 1), assume that there holds Then there exists a positive threshold Proof. The proof is essentially the same as the one in [13] with minor changes due to the fact that, in our case, (3.5) involves the whole integrand; see also [25,Lemma 6.2].
Combining (3.6), (3.7) and a standard covering argument, we obtain (3.8) and the proof is complete.
, with Hölder inequality we can rearrange (3.6) as follows: Let us point out a particularly helpful inequality contained in the proof of Lemma 3.1.
for c ≡ c(data p(·) ). Moreover, the following inequalities are satisfied: ) and for all σ ∈ 0, min σ g , q n − 1 , where σ g is the same higher integrability threshold appearing in Lemma 3.1.

3.2.
Proof of Theorem 1. The proof of Theorem 1 relies on the following result.
. Then, there exist a threshold radius R * ≡ R * (data) ∈ (0, 1] and a smallness parameter ε ≡ ε(data) For the sake of simplicity, we split the proof into six steps.
Step 1: Setting a threshold radius. As mentioned in Section 3.1, there is no loss of generality in reducing the size of the half ball we are working on. Precisely, in addition to (3.2), we choose a radius R ∈ (0, R * ], where now it is for σ 0 ∈ (0, 1) defined as In (3.22), σ g and σ ′ g are the higher integrability thresholds appearing Lemma 3.1, therefore, given Moreover, in addition to (3.3), another straightforward consequence of the restriction imposed in (3.21) yields that Hence, combining (3.23) and (3.24) we can conclude that Let us stress that by continuity, for any pointx ∈B + R for which p(x) ≥ n, we can find a small ball . Combining this information with (3.7) and observing that, by (3.22) with Sobolev-Morrey embedding theorem we obtain that u ∈ C 0, σ 0 . Therefore, for the rest of the paper, we shall assume that γ 2 < n. Moreover, since from now on we work on sets of the type B ̺ (x 0 ) ∩ B + R with x 0 ∈ B + R and ̺ ∈ (0, R − |x 0 |), we shall simplify the notation in (2.5) as follows: p 1 (x 0 , ̺) ≡ p 1 (̺) and p 2 (x 0 ; ̺) ≡ p 2 (̺).
Furthermore, h solves the Euler-Lagrange equation Notice that, by the results in [40] there holds that Recalling [13,Lemma 3.4], see also [37, Proof of Lemma 2] there holds that for all ϑ ∈ n 1 − p2(̺) q , n with c ≡ c(n, N, γ 1 , γ 2 , λ, Λ, q). For (3.39) we also used that, by and, recalling also (3.28), we see that for c ≡ c(data p(·) ). Merging the content of the two previous displays and proceeding as in the last part of Step 2 we end up with with c ≡ c(data). Collecting inequalities (3.35) and (3.41) we obtain where c ≡ c(data).
All in all, we have just proved that if Now, by the continuity of Lebesgue's integral and of the mapping x 0 → p 2 (x 0 , ̺), we can conclude that if (3.28) holds for x 0 on B ̺ (x 0 )∩B + R then it holds also on B ̺ (y)∩B + R for all y ∈B + 1 belonging to a sufficiently small, relatively open neighborhood of x 0 , say, B x0 ⊂B + R . Then the set is relatively open, so via (3.54) we can conclude that where c ≡ c(data, Dg L q (B + 1 ) ). By (3.55) and the well-known characterization of Hölder continuity due to Campanato and Meyers we can conclude thatũ is 1 − n q -Hölder continuous in a neighborhood of D 0 , which in turn implies that u ∈ C 0,1− n q loc (D 0 , M).

Full boundary regularity
In this section we recover a regularity criterion based on the result in Theorem 1. The main preliminary step consists in proving compactness of sequences of minimizers of (3.1) under uniform assumptions, see [9,13,47].
Remark 4.1. We will always assume that γ 2 < n, otherwise, as stressed in Step 1 of the proof of Theorem 1, we would have u Hölder continuous in a small neighborhood of any pointx ∈B + 1 so that p(x) ≥ n for free by Morrey's embedding theorem. and respectively. For each j ∈ N, let u j ∈ W 1,pj (·) (B + 1 , M) be a constrained minimizer of where the manifold M is as in (2.6) and the sequence {g j } ⊂ W 1,q (B + 1 , M), uniformly satisfying (2.7), is weakly convergent to some g 0 ∈ W 1,q (B + 1 , M). Then, there exists a subsequence, still denoted by {u j }, such that for someσ > 0 and any R ∈ (0, 1). In particular, u 0 is a constrained minimizer of the functional Finally, if x j is a singular point of u j and x j → x 0 , then x 0 is a singular point for u 0 .
Proof. For the reader's convenience, we split the proof into three steps.