On the Hölder continuity for a class of vectorial problems

1 Introduction

In this paper we establish Hölder regularity for vector-valued minimizers of a class of integral functionals of the Calculus of Variations. We shall apply such results to minimizers of quasiconvex integrands, therefore satisfying the natural condition to ensure existence in the vectorial setting.

For equations and scalar integrals, such a topic is strictly related to the celebrated De Giorgi result in [1]. Several generalizations in the scalar case have then been given, let us mention the contribution of Giaquinta-Giusti [2], establishing Hölder regularity for minima of non differentiable scalar functionals.

The question whether the previous theory and results extend to systems and vectorial integrals was solved in [3] by De Giorgi himself constructing an example of a second order linear elliptic system with solution x|x|y, y > 1 (see the nice survey [4]; we also refer to the paper [5] for the most recent result and an up-to-date bibliography on the subject). Motivated by the above mentioned counterexamples, in the mathematical literature there are two different research directions in the study of the regularity in the vector-valued setting: partial regularity as introduced by Morrey in [6], i.e., smoothness of solutions up to a set of zero Lebesgue measure, and everywhere regularity following Uhlenbeck [7]. For more exhaustive lists of references on such topics see for example [8, 9, 10].

Let us now introduce our working assumptions. Given n, N ≥ 2, and a bounded open set Ω ⊆ ℝⁿ, let f : Ω × ℝ^N×n → ℝ be a function such that there exist Carathéodory functions F_α : Ω × ℝⁿ → ℝ, α ∈ {1, …, N}, and G : Ω × ℝ^N×n → ℝ, such that for all ξ ∈ ℝ^N×n and for 𝓛ⁿ-a.e. x ∈ Ω

Here, we have adopted the notation

where ξ^α ∈ ℝⁿ, α ∈ {1, …, N}, is the α-th row of the matrix ξ.

Furthermore, we assume on each function F_α the following growth conditions: there exist an exponent p ∈ (1, n), constants k₁, k₂ > 0 and a non-negative function a ∈ Llocσ (Ω), σ > 1, such that for all α ∈ {1, …, N}, for all λ ∈ ℝⁿ and for 𝓛ⁿ-a.e. x ∈ Ω

In addition, we assume that G is rank-one convex and satisfies for all ξ ∈ ℝ^N×n and for 𝓛ⁿ-a.e. x ∈ Ω

for some q ∈ [1, p), and a non-negative function b ∈ Llocσ (Ω) (for the precise definition of rank-one convexity and other generalized convexity conditions see Section 2).

Consider the energy functional ℱ defined for every map u ∈ Wloc1,p (Ω, ℝ^N) and for every measurable subset E ⊂ ⊂ Ω by

The main result of the paper concerns the regularity of local minimizers of the functional ℱ. We recall for convenience that a function u ∈ Wloc1,p (Ω, ℝ^N) is a local minimizer of ℱ if for all φ ∈ W^1,p(Ω, ℝ^N) with supp φ ⋐ Ω

Existence of local minimizers for ℱ is not assured under the conditions of Theorem 1.1 since f might fail to be quasiconvex under the given assumptions. In the statement we have chosen to underline the only conditions needed to establish the regularity result. For the existence issue see [11, 12, 13]. Despite this, we shall give some non-trivial applications of Theorem 1.1 in Section 5. In particular, by using the function introduced by Zhang in [14], we construct examples of genuinely quasiconvex integrands f, which are not convex, and satisfying (1.1)-(1.4). Furthermore, by considering the well-know Šverák’s example [15], we exhibit an example of a convex energy density f satisfying the regularity assumptions with non-convex principal part F and with the perturbation G rank-one convex but not quasiconvex. For more examples see Section 5.

The special structure of the energy density f in (1.1) permits to prove Hölder regularity by applying the De Giorgi methods to each scalar component u^α of the minimizer u. More precisely, inspired by [16], we show that each component u^α satisfies a Caccioppoli type inequality, and then it is local Hölder continuous by applying the De Giorgi’s arguments; see [8, 17]. As regards the application of the techniques of De Giorgi in the vector-valued case but in a different framework we quote [18]; for related Hölder continuity results for systems we quote [19, 20, 21]. We remark that in [22] local y -Hölder continuity for every y ∈ (0, 1) has been proved for stationary points of similar variational integrals with rank-one convex lower order perturbations G differentiable at every point and with principal part F(ξ) = |ξ|^p.

In Section 4 we consider the case of polyconvex integrands. Precisely, the Hölder continuity of local minimizers is obtained under the same structural assumptions on F and suitable polyconvex lower order perturbations G depending only on the (N − 1) × (N − 1) minors of the gradient, see Theorem 4.1. The more rigid structure of the energy density f allows to obtain regularity results under weaker assumptions on the exponents when compared to Theorem 1.1, see Remark 4.2 and Example 1 in Section 5. We notice that in the recent papers [16, 23] the local boundedness of minimizers has been established for more general energy functionals ℱ with polyconvex integrands and under less restrictive conditions on the growth exponents.

We remark that the assumption p < n is not restrictive. Indeed, it is well-known that the regularity results still hold true if p ≥ n, even without assuming the special structure of f in (1.1). This is a consequence of the p-growth satisfied by f, the Sobolev embedding, if p > n, together with the higher integrability of the gradient if p = n (see [8, Theorem 6.7]).

We finally resume the contents of the paper. In Section 2 we introduce the various convexity notions in the vectorial setting of the Calculus of Variations and we recall De Giorgi’s regularity result. Section 3 is devoted to the proof of Theorem 1.1. In Section 4 we deal with functionals with a polyconvex lower order term G. Finally, in Section 5 we provide several non trivial examples of application of our regularity results.

2 Preliminaries

3 Proof of Theorem 1.1

The specific structure (1.1) of the energy density f yields a Caccioppoli inequality on every sub-/superlevel set for any component u^α of u. To provide the precise statement we introduce the following notation: given x₀ ∈ Ω, 0 < t < dist(x₀, ∂Ω), and with fixed k ∈ ℝ and α ∈ {1, …, N} set

The following lemma finds an important application in the hole-filling method. The proof can be found for example in [8, Lemma 6.1].

We are now ready to prove the local Hölder continuity of local minimizers.

4 The polyconvex case

In this section we deal with the case of a suitable class of polyconvex functions G. We will exploit their specific structure to obtain Hölder continuity results not included in Theorem 1.1. We shall use extensively the notation introduced in Section 2.1.

For u ∈ Wloc1,p (Ω; ℝ^N) and E ⊂ ⊂ Ω a measurable set, we shall consider functionals

with Carathéodory integrands f : Ω × ℝ^N×n → ℝ, n ≥ N ≥ 3, satisfying

We assume that the functions F_α are as in the previous section. In particular, we assume that

there exist p ∈ (1, n), k₁, k₂ > 0 and a non-negative function a ∈ Llocσ (Ω), σ > 1, such that

for all λ ∈ ℝⁿ and for 𝓛ⁿ-a.e. x ∈ Ω

As far as G : Ω × ℝ^N×n → ℝ is concerned, G depends only on (N − 1) × (N − 1) minors of ξ as follows:

For every α ∈ {1, ⋯, N} we assume that G_α : Ω × ℝ^N → ℝ is a Carathéodory function, λ ↦ G_α(x, λ) convex, such that

there exist r ∈ [1, p) and a non-negative function b ∈ Llocσ (Ω), σ > 1, such that

for all λ ∈ ℝ^N and for 𝓛ⁿ-a.e. x ∈ Ω.

The key result to establish Theorem 4.1 is, as in the case of Theorem 1.1, the following Caccioppoli’s type inequality which improves Proposition 3.1. We state it only for the first component u¹ of u. We recall that the super-(sub-)level sets are defined as in (3.1). Moreover, we use here the following notation:

For the sake of simplicity, in the Lebesgue norms we will avoid to indicate the target space of the functions involved.

We limit ourselves to exhibit the proof of Proposition 4.3, given that Theorem 4.1 follows with the same lines of the proof of Theorem 1.1.

5 Examples

We provide some non trivial applications of Theorems 1.1 and 4.1. In particular, we infer Hölder continuity of local minimizers to vectorial variational problems with quasiconvex integrands. The energy densities that we consider satisfy (1.1)-(1.4) and have neither radial nor quasi-diagonal structure. More in details, the integrands in Examples 1 and 2 are not convex, respectively they are polyconvex and quasiconvex, being F convex but G only polyconvex in the first case, and quasiconvex in the second. In Example 3 we construct a convex density though with non-convex principal part. Instead, the energy density f in Examples 4 and 5 is convex. In particular, in the first one F is convex and G is the rank-one convex non-quasiconvex function introduced by Šverák in [15]; in the second we construct a non-convex integrand F͠ by modifying F in Example 4, keeping the same G.

Being in all cases the resulting f quasiconvex, existence of local minimizers for the corresponding functional ℱ easily follows from the Direct Method of the Calculus of Variations.