REGULARITY OF MINIMIZERS FOR NONCONVEX VECTORIAL INTEGRALS WITH pq GROWTH VIA RELAXATION METHODS

Local Lipschitz continuity of local minimizers of vectorial integrals ∫Ω f(x,Du)dx is proved when f satisfies p-q growth condition and ξ↦f(x,ξ) is not convex. The uniform convexity and the radial structure condition with respect to the last variable are assumed only at infinity. In the proof, we use semicontinuity and relaxation results for functionals with nonstandard growth.


Introduction
In this paper, we study the regularity properties of the local minimizers of nonhomogeneous integral functionals of the form where Ω ⊂ R n is a bounded open set and Du denotes the gradient matrix of a vectorvalued function u : We say that a function u ∈ W 1,1 loc (Ω,R m ) is a local minimizer of I if f (x,Du) ∈ L 1 loc (Ω) and I(u,suppϕ) ≤ I(u + ϕ,suppϕ), (1.2) for any ϕ ∈ W 1,1 (Ω,R m ) with suppϕ Ω.
The main features of our functional I is the fact that the density energy f = f (x,ξ) is not convex with respect to ξ ∈ R nm and satisfies the so-called p-q growth condition, that is, there exist 1 < p < q and c 0 ,c 1 ,L > 0, such that (1. 3) The study is motivated by several problems in different contexts of mathematical physics as, for example, in nonlinear elasticity and homogenization.Most of the regularity results in the vectorial case m > 1 are obtained under the assumptions with g : Ω × [0,∞[ → [0,∞[ and f convex with respect to ξ.For example, in a recent paper, Cupini et al. [2] proved that if (1.3), with ξ ∈ R nm , and (1.4), with ξ ∈ R nm \ B R (0), hold and, in addition, f is p-uniformly convex at infinity, the local minimizers of I are Lipschitz continuous when 1 < p ≤ q < p(n + 1)/n.If q > p(n + 1)/n, there are some counterexamples to the regularity.See also Esposito et al. [6] and Mascolo and Migliorini [13] for related results.
The regularity of minimizers of nonconvex functionals is achieved, usually, via relaxation methods.For instance, Fonseca et al. [7], in scalar case m = 1 and in standard growth p = q, observed that, by the known relaxation theorems, local minimizers of nonconvex functionals are also minimizers of where f * * is the convex envelope of f with respect to ξ.Thus they can reduce to study the regularity of minimizers of convex functionals; see also Cupini and Migliorini [3].However, in the vectorial case, the relaxation methods are not so well clarified when dealing with p-q growth condition.
Here we consider nonhomogeneous density energies f = f (x,ξ), which are Caratheodory functions satisfying a local continuity condition with respect to x (see assumption (A2)).First, we prove a lower semicontinuity theorem when f is quasiconvex and q < pn/(n − 1) (see Theorem 2.2).Moreover, when as in [8], we are able to prove that (see Theorem 2.6) for all U Ω and u ∈ W 1,p (Ω,R m ), where (1.8) Our main regularity result follows by equality (1.7).

I. Benedetti and E. Mascolo 29
Let f be not convex in ξ, p-uniformly convex at infinity, and satisfy the conditions (1.4) and (1.6).Then, in Theorem 3.2, we prove that all local minimizers of I are local Lipschitz continuous.We give a sketch of the proof.
Let u be a local minimizer of I(u) and U Ω. Consider and the convex problem For classical results, (1.10) has at least one solution u and [2, Theorem 1.1] implies that u ∈ W 1,∞ loc (U,R m ).We prove that The proof of the last equalities is based on the representation formula (1.7), on a method introduced by De Giorgi [5], and on the related ideas contained in Fusco [9] and in Marcellini [12].Then, since (1.11) implies that u is a solution of the convex problem (1.10), we get u ∈ W 1,∞ loc (U,R m ).Moreover, we exhibit a class of density energies for which (1.6) is satisfied.Finally, we study the relationship between (1.6) and the Lavrentiev phenomenon.In particular, we show that under (1.4) and (1.6), we do not have the occurrence of the phenomenon.
In conclusion, we observe that, in the scalar case, the regularity of minimizers for nonconvex functionals is often the first step to prove the existence, see Mascolo and Schianchi [14] and Fonseca et al. [7], then it would be interesting to complete these researches in that direction.
This paper is organized as follows.In Section 2, we give the proof of the semicontinuity theorem and the characterization of Ᏺ p,q and Section 3 is devoted to the study of the regularity of local minimizers.

Semicontinuity and relaxation
Let Ω ⊂ R n be an open bounded set and let f : Ω × R nm → R be a nonnegative Caratheodory function satisfying the following assumptions: (A1) there exist q > 1 and L > 0 such that for almost everywhere x ∈ Ω and for every ξ ∈ R nm ; (A2) there exists a modulus of continuity λ(t) (i.e., λ(t) is a nonnegative increasing function that goes to zero as t → 0 + ) such that for every compact subset Ω 0 ⊂ Ω, there exists x 0 ∈ Ω 0 such that for all x ∈ Ω 0 and ξ ∈ R nm .
Let x ∈ Ω 0 be such that f (x,ξ) > f (x 0 ,ξ), then (2.2) can be written as Thus if δ is such that λ(δ) < 1, then for all Ω 0 Ω with diam(Ω 0 ) < δ, (2.3) holds with We say that a function f (x,ξ) is quasiconvex with respect to ξ in the Morrey's sense if for every ( First, we prove the following lower semicontinuity theorem in Sobolev spaces below the growth exponent for the density energy. Theorem 2.2.Let f be a quasiconvex function satisfying (A1) and (A2) and let 1 < p < q < n/(n − 1)p.Then ) Without loss of generality we may assume that Let Ω 0 be an open set compactly contained in Ω.For every integer ν such that λ(1/ν) < 1, we consider a subdivision of , and diamΩ i < 1/ν.By (A2) and Remark 2.1, there exists x i ∈ Ω i for which I. Benedetti and E. Mascolo 31 for all x ∈ Ω i and ξ ∈ R nm .Then by using the second inequality of (2.8) (2.9) We estimate the first integral in the right-hand side.Since q < pn/(n − 1), by the lower semicontinuity result for functionals with homogeneous density energy contained in [8,Theorem 4.1], we have Ωi The first inequality of (2.8) implies that (2.11) By (2.7) and (2.10), there exists M > 0 such that and consequently we obtain (2.13) Thus we go to the limit as ν → ∞ and then we get the result as Ω 0 ↑ Ω.
Suppose that f satisfies (A1) and (A2) and let Q f (x,ξ) be the quasiconvex envelope of f with respect to the second variable, that is, By the results contained in Dacorogna [4] and in Giusti [10], we have that (A1) implies By the definition of Q f , we have that (2.16) Now we show that when f satisfies (A2), Q f satisfies the same property.
, where g : R nm → R has q-growth: where L > 0 is a constant and a ∈ C 0,α (Ω).In this case, both f and Q f satisfy (A1) and We introduce the functional Ᏺ p,q : (2.23) The following theorem holds.
Proof.It is not difficult to check that the arguments in the proof of Theorem 2.2 can be applied to the quasiconvex functional Ω Q f (x,Du)dx by choosing a decomposition of taking the infimum over all such sequences, (2.24) follows.
When u ∈ W 1,q (Ω,R m ), by the standard relaxation results, there exists a sequence (w j ) ∈ W 1,q (Ω,R m ) such that w j u in W 1,q (Ω,R m ) and which easily implies that (2.24) is an equality.
Assume now a p-coercivity condition on f .(A1 ) There exist c 0 ,c 1 ,L > 0 such that (2.27) In the following, f * * (x,ξ) denotes the lower convex envelope of f with respect to the second variable.We are able now to give a characterization of Ᏺ p,q .
Theorem 2.6.Let f satisfy (A1 ) and (A2) and let 1 < p < q < p(n + 1)/n.Assume that ) Proof.Consider a smooth kernel ϕ ≥ 0 in R m with support in B(0,1), R n ϕ(x)dx = 1, and ϕ j (x) = j n ϕ( jx).For each j ∈ N, consider ϕ j * u ∈ W 1,q (U,R m ).Again for the standard relaxation results, we can select a sequence v jk ∈ W 1,q (U,R m ) such that lim k→∞ v jk = ϕ j * u weakly in W 1,q (U,R m ) and We may extract a diagonal subsequence u j = v jk( j) such that u j u in W 1,p (U,R m ) and (2.30) For every positive ν ∈ N with ν ≥ ν = inf{ν : λ(1/ν) < 1}, we consider a subdivision of U in the open sets U i such that i |U i | = |U|, U i U j = ∅, for i = j and diam(U i ) < 1/ν ∀i.
(2.39) Indeed for all u k ∈ W 1,p (U,R m ) with f (x,Du k ) ∈ L 1 (U) and u k u in W 1,p (Ω,R m ), we have Now we exhibit a class of integrands f : We say that f : For f : Ω × R nm → R, we define the rank-one convex envelope as R f (x,A) = sup{g ≤ f : g rank-one convex with respect to A}. (2.43) We prove the following result.
Proposition 2.8.Let Ω be a bounded open set of R n and let f : Assume that there exists a function g : for every x ∈ Ω and for every t > 0.Moreover, assume that there exists a measurable, nonnegative function α : Then, Proof.We use the same techniques contained in Dacorogna [4].First, we prove that g * * (x,|A|) ≥ f * * (x,|A|).Let x ∈ Ω and let ε be fixed, then by the characterization of the convex envelope, there exist λ ∈ [0,1] and b,c ∈ R + such that (2.48) From the arbitrariness of ε, we have the claimed result.To prove the reverse inequality, we first show that (2.45) implies that g * * (x,t) is not decreasing with respect to t. Fixing x ∈ Ω, we have in contradiction with the convexity of the function t → g * * (x,t).Now let v,w > 0 with v > w.If 0 < w < v < α(x), we have I. Benedetti and E. Mascolo 37 if 0 < w < α(x) < v, we have assume now that α(x) < w < v and g * * (x,v) < g * * (x,w), then there exists λ ∈ (0,1) such that w = λv + (1 − λ)α(x), so we get again in contradiction with the convexity of the function g * * .
Therefore, we have proved that g * * (x,|A|) = f * * (x,A).We are going now to prove that R f = f * * .

.61)
As a final remark, we point out that condition (1.6), that is, Q f (x,ξ) = f * * (x,ξ), is connected with the Lavrentiev phenomenon.In an abstract framework, let X and Y be two topological spaces of weakly differentiable functions, with Y ⊂ X and Y dense in X.We say that there is the Lavrentiev phenomenon when (2.62) The following result holds.

Relaxation and regularity
In this section, we apply the relaxation equality contained in Theorem 2.6 to get the W 1,∞ regularity for local minimizers of nonconvex functionals.
Let f : Ω × R nm → R be a Caratheodory function satisfying (A1) and (A2) and the following assumptions: (B1) there exist R > 0 and a function g such that, for almost everywhere x ∈ Ω and every (B2) f is p-uniformly convex at infinity, with p ≤ q, that is, there exist p > 1 and ν > 0 such that, for almost everywhere x ∈ Ω and for every ξ 1 ,ξ 2 ∈ R nm \ B R (0) endpoints of a segment contained in the complement of B R (0), I. Benedetti and E. Mascolo 39 (B3) for almost everywhere x ∈ Ω and every ξ ∈ R nm \ B R (0), let D + t g(x,|ξ|) be the right-side derivative of g with respect to t and denote Then for every ξ ∈ R nm \ B R (0), the vector field x → D + ξ f (x,ξ) is weakly differentiable and where c = c(n, p, q,L,L 1 ,R,ν) and β = β(n, p, q).
In the sequel, we need some properties of functions satisfying (A1), (A2), and the p-uniformly convexity at infinity.In [2, Lemma 2.2], it is proved that assumption (B2) implies that f is p-coercive, that is, there exist c 0 ,c 1 > 0 such that By [2, Theorem 2.5(iv)], it follows that if f satisfies the condition (B2), there exists R 0 depending on ν, p, q, L such that for almost everywhere x ∈ Ω and ξ ∈ R nm \ B R0 (0).Moreover, by assumption (A2) and Remark 2.1, it is easy to check that there exists δ > 0 such that, for every ξ ∈ B R0 (0) and x ∈ U, with U Ω and diamU < δ, there exists then, for every ξ 1 ,ξ 2 ∈ B R0 (0), where c, c 2 are positive constants.
40 Regularity of minimizers for nonconvex integrals We now consider ξ 1 ,ξ 2 ∈ R nm \ B R0 (0).By (3.8) and the growth condition, we obtain with c 3 = c 3 (L, q).Therefore, it is easy to check that there exist c 2 = c 2 (R 0 ,U) and c 3 (L, q) such that Finally, for every x ∈ U, where U Ω, and for every ξ ∈ R nm , (3.8) implies that for all t ∈ [0,1], where c 4 depends on ν and R 0 .
Our main result is the following regularity theorem.
.5) Cupini, et al. in [2, Theorem 1.1] proved the following regularity result for the local minimizer of I when f is convex with respect to ξ.