On the Hölder continuity for a class of vectorial problems

: In this paper we prove local Hölder continuity of vectorial local minimizers of special classes of integral functionals with rank-one and polyconvex integrands. The energy densities satisfy suitable structure assumptions and may have neither radial nor quasi-diagonal structure. The regularity of minimizers is obtained by proving that each component stays in a suitable De Giorgi class and, from this, we conclude about the Hölder continuity. In the final section, we provide some non-trivial applications of our results.


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 di erentiable 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| γ , γ > (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 di erent 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].
(1. 3) In addition, we assume that G is rank-one convex and satis es for all ξ ∈ R N×n and for L n -a.e. x ∈ Ω |G(x, ξ )| ≤ k |ξ | q + b(x) (1.4) for some q ∈ [ , p), and a non-negative function b ∈ L σ loc (Ω) (for the precise de nition of rank-one convexity and other generalized convexity conditions see Section 2).
Consider the energy functional F de ned for every map u ∈ W ,p loc (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 ∈ W ,p loc (Ω, R N ) is a local minimizer of F if for all φ ∈ W ,p (Ω, R N ) with supp φ Ω F (u; supp φ) ≤ F (u + φ; supp φ). 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 satis es 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 di erent framework we quote [18]; for related Hölder continuity results for systems we quote [19][20][21]. We remark that in [22] local γ-Hölder continuity for every γ ∈ ( , ) has been proved for stationary points of similar variational integrals with rank-one convex lower order perturbations G di erentiable 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 − ) × (N − ) minors of the gradient, see Theorem 4.1. The more rigid Brought to you by | Universita di Bologna -Area Biblioteche e Servizi allo Studio Authenticated Download Date | 3/11/20 1:13 PM 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 satis ed 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 nally 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.

Preliminaries . Convexity conditions
Motivated by applications to nonlinear elasticity, J. Ball in 1977 pointed out in [11] that convexity of the energy density is an unrealistic assumption in the vectorial case. Indeed, it con icts, for instance, with the natural requirement of frame-indi erence for the elastic energy. Then, in the vector-valued setting N > , di erent convexity notions with respect to the gradient variable ξ play an important role. We recall all of them in what follows.
for every matrix ξ ∈ R N×n , where adj i ξ is the adjugate matrix of order i ∈ { , . . . , N ∧ n} of ξ , that is the and that in the scalar case all these notions are equivalent (see for instance [13,Theorem 5.3]).
On the other hand, none of the previous implications can be reversed except for some particular cases. We refer to [13,Chapter 5] for several examples and counterexamples. In particular, in Section 5 we shall extensively deal with Šverák's celebrated counterexample to the reverse of the last implication above.

. De Giorgi classes
In this section we recall the well-known regularity result in the scalar case due to De Giorgi [1].

Theorem 2.3.
Let v ∈ DG(Ω, p, γ, γ * , δ) and τ ∈ ( , ). There exists a constant C > depending only upon the data and independent of v, such that for every pair of balls moreover, there existsα ∈ ( , ) depending only upon the data and independent of v, such that

Proof of Theorem 1.1
The speci c 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 Proof. Without loss of generality we may assume α = . For the sake of notational convenience we drop the x -dependence in the notation of the corresponding sub-/superlevel set. We start o with proving the inequality on the super-level sets.
Then notice that L n -a.e. in {η > } ∩ {u > k} Thus, since Du − A is a rank-one matrix, the rank-one-convexity of G yields By the local minimality of u, (3.4), (3.7) and taking into account that L n -a.e. in Ω The latter inequality and (3.4) imply that (3.8) (3.6), the convexity of t → |t| p and (3.3), we get We now estimate the last integral at the right hand side. The growth condition in (1.4) for G, Hölder's and Young's inequalities imply, for some c = c(k , p, q) > , Hence, by taking into account estimates (1.3), (3.9) and (3.10), we obtain   (3.2) follows at once from (3.11), by taking into account that L n (A k,R ) ≤ L n (B R ) ≤ .
In conclusion, the analogous estimate with B k,R in place of A k,R follows from (3.2) itself since −u is a local minimizer of the integral functional with energy densityf (x, ξ ) := f (x, −ξ ).
The following lemma nds an important application in the hole-lling method. The proof can be found for example in [ We are now ready to prove the local Hölder continuity of local minimizers.

The polyconvex case
In this section we deal with the case of a suitable class of polyconvex functions G. We will exploit their speci c 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 ∈ W ,p loc (Ω; R N ) and E ⊂⊂ Ω a measurable set, we shall consider functionals with Carathéodory integrands f : We assume that the functions Fα are as in the previous section. In particular, we assume that there exist p ∈ ( , n), k , k > and a non-negative function a ∈ L σ loc (Ω), σ > , such that for all λ ∈ R n and for L n -a.e. x ∈ Ω As far as G : Ω × R N×n → R is concerned, G depends only on (N − ) × (N − ) minors of ξ as follows: For every α ∈ { , · · · , N} we assume that Gα : Ω × R N → R is a Carathéodory function, λ → Gα(x, λ) convex, such that there exist r ∈ [ , p) and a non-negative function b ∈ L σ loc (Ω), σ > , such that for all λ ∈ R N and for L n -a.e. x ∈ Ω.
is a polyconvex function, satisfying 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 rst component u of u. We recall that the super-(sub-)level sets are de ned as in (3.1). Moreover, we use here the following notation: u := (u , u , · · · , u N ).
For the sake of simplicity, in the Lebesgue norms we will avoid to indicate the target space of the functions involved. (

Proposition 4.3 (Caccioppoli inequality on sub-/superlevel sets). Let f be as in
The same inequality holds substituting A k,R,x with B k,R,x .
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.

Proof of Proposition 4.3.
The proof is the same of that of Proposition 3.1 up to inequality (3.9). By keeping the notation introduced there, (3.9) and the left inequality in (4.2) imply with c depending on p, k . We exploit next the speci c structure of G. Taking into account the de nition of Using the growth condition (4.4), that in particular implies that Gα is non-negative, we get with c depending on k . Denoteû := (u , u , · · · , u N ). For every α ∈ { , · · · , N} we have with c depending on n and N. Since r < p we can use the Young's inequality with exponents p r and p p−r , so we have, L n -a.e. in A k,t ∩ {η > }, N, p, r).
Collecting the above inequalities, we get  c(k , n, N, p, r) > . By (4.6) (N− )r p−r < therefore by Hölder's inequality we get Analogously, Therefore by (4.9) and (4.10) we get with c = c(n, N, p, k , r) > . We now proceed as in the proof of Proposition 3.2: adding to both sides of (4.11) c A k,s |Du | p dx and using Lemma 3.2 we obtain that with ϑ as in (4.8) and c = c(n, N, p, k , k , r) > .
In conclusion, the analogous estimate with B k,t in place of A k,t follows from (4.7) itself since −u is a local minimizer of the integral functional with energy densityf (x, ξ ) := f (x, −ξ ).

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 rst 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 rst 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 F 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.

Example 1
Let n = N = and consider f : with p ≥ and r ≥ . We recall that, for all ξ ∈ R × , adj ξ ∈ R × denotes the adjugate matrix of ξ of order , whose components are We claim that f is a polyconvex, non-convex function satisfying the structure condition (4.1) with suitable Fα and G satisfying the growth conditions (4.2) and (4.4), respectively.
As far as the stucture is concerned, it is easy to see that (1.1) holds, if we de ne, for all α ∈ { , , } and The polyconvexity of f follows from the convexity of F and h (the latter holds true since r ≥ ), see e.g. [13].
Let us now prove that f is not convex. Consider the matrices ξ := εξ and ξ := −ξ , ε > , wherẽ We shall prove that for ε > su ciently small thus establishing the claim. Indeed, on one hand the right hand side rewrites as while on the other hand the left hand side rewrites as Note that φ ∈ C ((− , )) since p > . Simple computations show that φ ( ) = and φ ( ) = −r r < . Thus, for some δ ∈ ( , ) and for all ε ∈ ( , δ) we have φ (ε) < φ ( ) = . Thus φ(ε) < φ( ), and inequality (5.1) follows at once. By using Theorem 4.1 we have that, if p ∈ ( + √ ), and r ∈ , p +p , then the W ,p loc -local minimizers of the corresponding functional F are locally Hölder continuous.
We note that the arguments in [

Example 2
Let n, N ≥ . Given two matrices ξ , ξ in R N×n such that rank(ξ − ξ ) > , de ne Denoting Qdist(·, K) the quasiconvex envelope of the distance function from K, we de ne, for q ≥ , the quasiconvex function G : R N×n → [ , +∞) by where coK is the convex envelope of the set K. For all ϱ ∈ [ , ] de ne the energy density fϱ : R N×n → R, and note that fϱ satis es (1.1)-(1.4) and it is quasiconvex.
We claim that, xed p ≥ , there exists ϱ > such that, for every ϱ ∈ ( , ϱ ), fϱ is quasiconvex, but not convex. Given this for granted, by Theorem 1.1 we have that the W ,p loc -local minimizers of the corresponding functional F are locally Hölder continuous provided that ≤ q < p n . To prove the claim, we rst observe that the function G is not convex, since G − ((−∞, ]) turns out to be the set K, which is non-empty and non-convex. Indeed, by [14, Theorem 1.1, Example 4.3], the zero set of the quasiconvex function with linear growth ξ → Qdist(ξ , K) is K. This implies G − ((−∞, ]) = K.

Example 3
We give an example of an overall convex function f having non-convex principal part and convex lower order term.
Let ≤ q < p < n, µ > , and B := {z ∈ R n : |z| < }. With this aim we rst compute the Hessian matrices of Fα and F. Simple computations yield for all λ, ζ ∈ R n .
Let > and set G(ξ ) := ( + |ξ | ) q , and recall that for all ξ , η ∈ R N×n (cf. (5.4)) being q ≥ To show that f := F + G is convex we compute its Hessian, being clearly f ∈ C (R N×n ). We have In conclusion, since f satis es (1.1)-(1.4), its convexity assures the existence of W ,p loc -local minimizers of the corresponding functional F , which, in view of Theorem 1.1, are locally Hölder continuous.

Example 4
In what follows we construct an example of a convex energy density f satisfying (1.1)-(1.4) with G rank-one convex but not quasiconvex. With this aim we recall next the construction of Šverák's celebrated example in [15] in some details, following the presentation given in the book [13]. With this aim consider (5.6) and let h : L → R be given by Let P : R × → L be de ned as and set gε,γ(ζ ) := h(P(ζ )) + ε|ζ | + ε|ζ | + γ|ζ − P(ζ )| .
To prove the claim, since f ∈ C (R N×n ) we shall compute its Hessian. First note that F(ξ ) := N α= Fα(ξ α ) is uniformly convex on R N×n in view of (5.4), which, together with (5.8), yields for all ξ ∈ R N×n such that |π(ξ )| ≥ ϑ+ ε and for all η ∈ R N×n . In addition, using again that p > , by (5.7) we have Hence, the Hessian of f at ξ with |π(ξ )| < ϑ+ ε is non-negative provided that

Example 5
Finally, we give an example that exploits the full strength of the assumptions of Theorem 1.1 on the leading term. By keeping the notation introduced in Example 4, we shall modify F there to get a non-convex function so that the resulting principal term F is non-convex. On the other hand, the sum F + Gε turns out to be convex exploiting the uniform convexity of Gε on the subspace L for large values of the variable π(ξ ) (cf. (5.6) and (5.8)).