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

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


Introduction
In this paper we prove C 1,β -partial regularity for manifold constrained p(x)-harmonic maps. More precisely, we consider local minimizers of the functional , Ω ⊂ R n is an open subset and M ⋐ R N is an m-dimensional compact submanifold endowed with a suitable topology. We refer to Section 1 below for the precise notation. Our final outcome is that there exists a relatively open set Ω 0 ⊂ Ω of full measure such that u ∈ C 1,β (Ω 0 , M) and Σ 0 (u) = Ω \ Ω 0 has Hausdorff dimension at the most n − γ 1 , see Section 1 for the assumptions and the relevant definitions.
Moreover, after imposing some extra restrictions on the variable exponent p(·), we are able to provide a further reduction to the Hausdorff dimension of the singular set of M-constrained minimizers of the p(x)energy Let us now put our results into the context of the available literature. Functionals with a variable growth exponent modelled on the one in (0.2) have been introduced, in the setting of the Calculus of Variations and Homogenization, in the fundamental work of Zhikov [47,48,49,50]. Energies as in (0.2) also occur in the modeling of electro-rheological fluids, a class of non-newtonian fluids whose viscosity properties are influenced by the presence of external electromagnetic fields [3,39]. As for regularity, the first result in the vectorial case has been obtained Coscia & Mingione [8], who proved that local minimizers of the functional (0.2) are locally C 1,β -regular in the unconstrained case. This is the optimal generalization of the classical results of Uhlenbeck concerning standard case when p(x) is a constant. We refer to [28,29,32,35,44,45] for a survey of regularity results in the standard 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 [36,37,38,42,43], where the authors have established partial regularity results for unconstrained minimizers that are on the other hand obviously related to the constrained case. In particular, in [42] Tachikawa gives an interesting partial regularity result and a singular set estimates for a class of functionals related to the constrained minimization problem in which the minimizer is assumed to take values in a single chart. This generalizes the well-known results of Giaquinta & Giusti [19] 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 suitably topological assumptions are considered on the manifold M and optimal regularity conditions are in force on p(·) and k(·). Our first main result is the following: Theorem 1 Let u ∈ W 1,p(·) loc (Ω, M) be a local minimizer of the functional in (0.1), where p satisfies assumptions (P1)-(P2), k satisfies (K1)-(K2) and M is as in (M1)-(M2) below. Then there exists a relatively open set Ω 0 ⊂ Ω such that u ∈ C 1,β (Ω 0 , M) and H n−γ 1 (Ω \ Ω 0 ) = 0.
By considering further assumptions on the variable exponent p(·) we are then able to prove a better dimension estimate for the singular set. This is in the following: ii. if n > [γ 1 ] + 1, then the Hausdorff dimension of the singular set is at the most n − [γ 1 ] − 1.
As they are stated, our results are the natural generalization of the classical ones in [23,30,40] for the case p(x) ≡ constant. For the vectorial quasiconvex case with standard p-growth we refer to the recent work of Hopper [27]. The extension we make here to the variable exponent case requires a number of non-trivial additional ideas and tools, especially, as far as the dimension estimates stated in Theorem 2 are concerned. This is also related to the recent, aforementioned paper of Tachikawa [42], 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 functional with (p, q)-growth (following the terminology introduced by Marcellini [33,34], see also [14]). These are integral functionals of the type w → F(x, Dw) dx, where the integrand F(·) satisfies The study of such functionals has undergone an intensive development over the last years, see for instance [5,10,31,33,34,35]. Another prominent model in this class is the so called double phase energy, where it is This model shares several features with the variable growth exponent and has been again introduced by Zhikov [49]. Indeed, here once again the growth exponent with respect to the gradient variable is determined by the space variable x. Indeed, the growth exponent changes according to the the positivity of the coefficient a(x). There are several analogies between the variable exponent energy and the double phase one.
In particular, one should compare the use of Gehring's lemma based reverse Hölder inequalities made here and the reverse Hölder inequality coming from fractional differentiability made in [6,7]. Moreover, compare the use of localization methods based on p-harmonic type approximation made 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.

Notation, main assumptions and functional setting
Here we establish some basic notation and display the precise assumption we are going to work with. In the following Ω ⊂ R n will denote an open, bounded subset of R n , n ≥ 2. We denote by B r (x 0 ) = x ∈ R n : |x − x 0 | < r the open ball with center x 0 and radius r > 0; when not important or clear from the context, we simply write: B r ≡ B r (x 0 ). With U ⊂ R n being a measurable subset with positive, finite Lebesgue's measure ∞ > |U| > 0 and with f : U → R N being a measurable map, we denote by the integral average of f over U. In particular, when U = B r , we will indicate only the radius and, if necessary, the centre of the ball, i. e.: When considering the functional in (0.1) the exponent p(·) and the coefficient k(·) will always satisfy respectively.
For a given B r ⋐ Ω we denote Moreover we need to impose some restriction on the manifold M. Precisely, we shall assume that Here [x] denotes the integer part of x. Clearly, assumption (M2) requires that γ 2 < m. First, in the unconstrained case, we introduce the classical Orlicz spaces, which are based on the finiteness of a certain energy, defined through an N function. We clarify this in the following definition.

Remark 2
In order to extrapolate good regularity properties for minimizers of functionals with ϕ growth, we need to assume something more. Precisely, from now on, in addition to the basic assumptions listed in Definition 1 we will also suppose that ϕ ∈ C 1 [0, ∞) ∩ C 2 (0, ∞) and that ϕ ′ (t) ∼ tϕ ′′ (t), where the constant implicit in "∼" depends only on the characteristics of ϕ. This is equivalent to the so-called ∆ 2 condition, since t → ϕ(t) is non decreasing, see [10].

Definition 2 Let ϕ be an N function in the sense of Definition 1 and Remark 2. Given an open
and, consequently, The definitions of the variants W 1,ϕ 0 (Ω, R N ) and W 1,ϕ loc (Ω, R N ) come in an obvious way from the one of W 1,ϕ (Ω, R N ).
A generalization of the Orlicz spaces are the so-called Musielak-Orlicz spaces, i.e., spaces of functions characterized by the finiteness of an energy defined by a generalized Young function ϕ = ϕ(x, t), see [25] for the definition and relevant properties. For the sake of clarity, we shall consider ϕ(x, t) = t p(x) .

Definition 3 Given an open
and, consequently, The variants W 1,p(·) 0 (Ω, R N ) and W 1,p(·) loc (Ω, R N ) are defined in an obvious way. It is well known that, under assumption (P1), the set of smooth maps is dense in W 1,p(·) (Ω, R N ), see e. g. [14,47,49,50]. Since we are dealing with maps taking values in manifolds, following [9,27] we recall the definition of the main function spaces in which we set our problem.
Owing to the p(x)-growth behavior of our integrand, we display our definition of local minimizer.

Preliminary regularity results
Here we collect some well known results in the framework of regularity. We start with two results concerning some reference estimate for unconstrained ϕ-harmonic maps.
The next one is the ϕ-harmonic approximation lemma, which will be crucial in our proof of partial regularity. Before stating it, we briefly recall what a ϕ-harmonic map is.
Lemma 2 [11] Let Ω ⊂ R n be an open subset and ϕ be an N-function in the sense of Definition 1 and Remark 2. For eveyθ > 0 andd ∈ (0, 1) there existsδ > 0 depending only onδ,d and on the characteristics of ϕ such that the following holds. Let B r ⊂ R n be a ball andB r denote either B r or B 2r .
We will use Lemma 2 with ϕ(t) = k 0 t p for some γ 2 ≥ p ≥ γ 1 > 1 and k 0 ∈ [λ, Λ], so the constants depending on the characteristics of ϕ actually depend on λ, Λ, γ 1 and γ 2 . We conclude our list with a couple of simple, but useful lemmas, which will be used several times in the forthcoming estimates.

Extensions
We recall some results concerning locally Lipschitz retractions. They have been extensively used in the realm of functionals with p-growth, see e. g.: [24,27]. 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. We use those theorems to construct suitable comparison maps which will be crucial in some steps of our proof. Proof. See e. g., [24] for the original proof, or [27] for a simplified version relying on some Lipschitz extension properties of maps between Riemannian manifolds.
Given any u ∈ W 1,p(·) loc (Ω, R N ), we are going to apply Lemma 6 to assure a local control on the L p(·) -norm of the gradient of a suitable projected image of u in terms of the L p(·) -norm of u itself. This is the content of the next lemma.
By a change of variables, the definition of the dual skeleton, the fact that M is ([ with c = c(N, M, γ 2 ) > 0. Now, for a sufficiently small 0 < ρ < min σ 2 , dist(M,∂Q) 2 and a point a ∈ B N ρ = b ∈ R N : |b| < ρ denote the translations Q a = b + a : b ∈ Q and X a = {b + a : b ∈ X}, so that one can define the retraction P a : Q a \ X a → M given by P a (b) = P(b− a). Then, by the chain rule, Fubini's theorem, (3.1) and (3.2) we obtain where c = c(N, M, γ 2 ). Estimate (3.3) and Markov's inequality then render the existence of a positive c = c(N, M, γ 2 ) and aã ∈ B N ρ so that Lemma 7 will be particularly helpful when U is any ball B r or an annulus B r \ B ρ for a proper choice of r and ρ.

Partial regularity
We start by collecting some results which are by now well-known in the unconstrained case, see [3,8,13,47,49].
Proof. The proof is essentially contained in [13], Theorem 3.1 and it is valid for any map v ∈ W 1,p(·) loc (Ω, R N ), so it transfers verbatim for functions in W 1,p(·) loc (Ω, M). However, since we are dealing with bounded maps, we present a simplified proof of it including also the case in which the domain is an annulus A rθ = B r \B r(1−θ) for some 0 < θ < 1. Fix B r ⋐ Ω, r ∈ (0, 1). Using the Hölder continuity of x → p(x), the definition of p 1 (r), p 2 (r), γ 1 and γ 2 and the classical Poincaré's inequality for p = p 1 (r) we obtain Lemma 9 (Intrinsic Poincaré's inequality) Let p ∈ (1, ∞) and w ∈ W 1,p (Ω, R N ). Then for any γ ∈ max 1 p , n n+p , 1 there exists a positive c = c(n, N, p, γ) and exponents d 1 and d 2 such that holds whenever B r ⋐ Ω is such that r ≤ 1. Specifically, d 1 = nγ n−γp > 1 and d 2 = γ.
Proof. Fix B r ⋐ Ω with r ∈ (0, 1) and recall that, owing to the fact that 0 Apply the standard Sobolev-Poincaré's inequality with q = γp and q * = npγ n−γp and recall that r < 1 to get N, p, γ). This is what we wanted with d 1 = nγ n−γp > 1 and d 2 = γ < 1, by our choice of γ.

Remark 4
Since M is compact, for functions v taking values in M the dependence of the constants appearing in the inequalities in Lemmas 8 and 10 on v ∞ will be expressed as a dependence on M.
The next step is proving a higher integrability result for local minimizers of (0.1).
Lemma 11 Let u ∈ W 1,p(·) loc (Ω, M) be a local minimizer of (0.1) and B r ⋐ Ω be any ball. Then, Proof. Once Lemma 10 is available the proof is the same as the one in [47,49], see also [13]. Anyway, since due to the compactness of M we are dealing with bounded maps, we can present a very simple proof of this lemma. By Lemma 10 the Hölder continuity of p(·) and Lemma 9 with p = p 1 (r) we get that [p] 0,α , α) and d 2 < 1 is as in Lemma 9. The thesis follows from an application of Gehring's Lemma, [22], Chapter 6.
Remark 5 For later uses we can rewrite (4.8) by obvious means, as where c has the dependencies outlined in Corollary 12.
The next Corollary allows recovering some useful estimates for the average of the gradient of solutions to problem (0.1).
We are now ready for showing Theorem 1.

Proof of Theorem 1.
For the reader's convenience, we split the proof into three parts.

Remark 6
Given the features of f displayed in (4.51) and the estimates performed in the proof of Lemma 13, the Hölder continuity exponent β does not depend either onΩ, nor on f itself.

Dimension reduction and Monotonicity
In this section we obtain a further reduction of the dimension of the singular set of p(x)-harmonic maps, for p(·) ≥ 2 Lipschitz, thus improving, at least in this "special" case, the result given in Theorem 1. The proof essentially goes in two moments: first we show a lemma concerning the compactness of sequences of minimizers of (0.1), then, by strengthening the initial assumptions we can conclude by exploiting the monotonicity of a certain quantity, strictly related to the p(x)-energy. Those arguments are quite classical, see e. g. [23,42].
Lemma 14 (Compactness) Let (k j ) j∈N , (p j ) j∈N be two sequences of Hölder continuous functions satisfying respectively. For each j ∈ N, let u j ∈ W 1,p j (·) (B 1 , M) be a constrained local minimizer of where M is as in (M1)-(M2). Then, there exists a subsequence, still denoted by (u j ) j∈N , such that for some ̟ > 0 and any r ∈ (0, 1) and v is a constrained local minimizer of the functional for all r ∈ (0, 1). Finally, if x j is a singular point of u j and x j →x, thenx is a singular point for v.
Proof. The proof is divided into three steps.

Remark 7
We stress that Lemma 14 holds with p(·) ≥ γ 1 > 1 Hölder continuous rather than Lipschitz. We need stronger assumptions only to prove a suitable monotonicity formula.
The following is the monotonicity lemma, which relies on some modifications of the arguments in [16,19,42].
Now we can proceed as in [16,19,42] to further reduce the dimension of the singular set of p(x)-harmonic maps.
Proof of Theorem 2. Let δ 0 be as in Lemma 11,and set Then, by Lemma 11, u ∈ W 1,n+δ 0 /2 (Ω 1 , M), so, by Sobolev-Morrey's embedding theorem we obtain u ∈ C 0,σ (Ω 1 , M) with σ = 1 − n/(n + δ 0 /2). Then, given the Hölder continuity of u we can first move to a single chart and consider u as a solution to an unconstrained variational problem having features similar to the initial one, see (4.52). Then we may apply Lemma 13 to show the Hölder continuity of the gradient. The above observations stresses that it is enough to prove Theorem 2 on the subset Ω 2 = x ∈ Ω : p(x) < n , where Lemma 15 works.