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Abstract and Applied Analysis growth via relaxation methods
growth via relaxation methods
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2004
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2004
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english
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Abstract and Applied Analysis
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10.1155/s1085337504310079
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REGULARITY OF MINIMIZERS FOR NONCONVEX VECTORIAL INTEGRALS WITH pq GROWTH VIA RELAXATION METHODS IRENE BENEDETTI AND ELVIRA MASCOLO Received 10 January 2003 Local Lipschitz continuity of local minimizers of vectorial integrals Ω f (x,Du)dx is proved when f satisfies pq 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. 1. Introduction In this paper, we study the regularity properties of the local minimizers of nonhomogeneous integral functionals of the form I(u,Ω) = Ω f (x,Du)dx, (1.1) where Ω ⊂ Rn is a bounded open set and Du denotes the gradient matrix of a vectorvalued function u : Ω → Rm , n,m > 1. 1,1 (Ω, Rm ) is a local minimizer of I if f (x,Du) ∈ L1loc (Ω) We say that a function u ∈ Wloc and I(u,suppϕ) ≤ I(u + ϕ,suppϕ), (1.2) for any ϕ ∈ W 1,1 (Ω, Rm ) with suppϕΩ. 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 ∈ Rnm and satisfies the socalled pq growth condition, that is, there exist 1 < p < q and c0 ,c1 ,L > 0, such that c1 ξ  p − c0 ≤ f (x,ξ) ≤ L 1 + ξ q . (1.3) The regularity of minimizers in the pq growth condition (1.3) has been extensively studied in the last years, starting from some papers of Marcellini, see for example [11, 12]. Copyright © 2004 Hindawi Publishing Corporation Abstract and Applied Analysis 2004:1 (2004) 27–44 2000 Mathematics Subject Classification: 49N60 URL: http://dx.doi.org/10.1155/S1085337504310079 28 Regularity of minimizers for nonconvex integrals 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 f (x,ξ) = g x, ξ  x ∈ Ω, ξ ∈ Rnm , (1.; 4) 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 ∈ Rnm , and (1.4), with ξ ∈ Rnm \ BR (0), hold and, in addition, f is puniformly 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 I ∗∗ (u,Ω) = Ω f ∗∗ (x,Du)dx, (1.5) 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 pq growth condition. The lower semicontinuity with respect to the weak topology W 1,p (Ω, Rm ) of quasiconvex functional, with integrand satisfying (1.3), has been studied by Marcellini [12] for f = f (x,ξ) and q < p(n + 1)/n and by Fonseca and Malỳ [8] when f = f (ξ) and q < pn/(n − 1). Further in [8], a representation of the relaxed functional is given when Q f (ξ) = f ∗∗ (ξ), where Q f (ξ) denotes the quasiconvex envelope of f . 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 Q f (x,ξ) = f ∗∗ (x,ξ), (1.6) as in [8], we are able to prove that (see Theorem 2.6) for all U Ω and u ∈ W 1,p (Ω, Rm ), Ᏺ p,q (u,U) = I ∗∗ (u,U), where Ᏺ (u,Ω) = inf liminf p,q uk k→∞ Ω (1.7) f x,Duk dx, uk ∈ W 1,q Ω, Rm , uk u in W 1,p Ω, Rm Our main regularity result follows by equality (1.7). (1.8) . I. Benedetti and E. Mascolo 29 Let f be not convex in ξ, puniformly 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 1,p I(u,U) = inf I(v,U) : v ∈ W0 U, Rm + u (1.9) and the convex problem 1,p inf I ∗∗ (v,U) : v ∈ W0 U, Rm + u . (1.10) For classical results, (1.10) has at least one solution u and [2, Theorem 1.1] implies that 1,∞ u ∈ Wloc (U, Rm ). We prove that I(u,U) = I ∗∗ (u,U) = I ∗∗ (u,U). (1.11) 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,∞ (U, Rm ). (1.10), we get u ∈ Wloc 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. 2. Semicontinuity and relaxation Let Ω ⊂ Rn be an open bounded set and let f : Ω × Rnm → R be a nonnegative Caratheodory function satisfying the following assumptions: (A1) there exist q > 1 and L > 0 such that 0 ≤ f (x,ξ) ≤ L 1 + ξ q , (2.1) for almost everywhere x ∈ Ω and for every ξ ∈ Rnm ; (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 Ω0 ⊂ Ω, there exists x0 ∈ Ω0 such that f x0 ,ξ − f (x,ξ) ≤ λ x − x0 for all x ∈ Ω0 and ξ ∈ Rnm . 1 + f (x,ξ) , (2.2) 30 Regularity of minimizers for nonconvex integrals Remark 2.1. It is easy to check that (2.2) implies that there exists δ > 0, with λ(δ) < 1, such that when Ω0 ⊂ Ω is a compact set with diam(Ω0 ) < δ, there exists a modulus of continuity µ(t) in [0,δ[ such that f x0 ,ξ − f (x,ξ) ≤ µ x − x0 1 + f x0 ,ξ , (2.3) for all x ∈ Ω0 and for all ξ ∈ Rnm . In fact, fixing ξ ∈ Rnm for all x ∈ Ω such that f (x0 ,ξ) ≥ f (x,ξ), the condition (2.2) implies (2.3) with µ = λ. Let x ∈ Ω0 be such that f (x,ξ) > f (x0 ,ξ), then (2.2) can be written as f x0 ,ξ ≥ f (x,ξ) 1 − λ x − x0 −λ x − x0 . (2.4) Thus if δ is such that λ(δ) < 1, then for all Ω0 Ω with diam(Ω0 ) < δ, (2.3) holds with µ(t) = λ(t)/(1 − λ(t)). We say that a function f (x,ξ) is quasiconvex with respect to ξ in the Morrey’s sense if for every x ∈ Ω and ξ ∈ Rnm , Ω f x,ξ + Dϕ(y) d y ≥ Ω f (x,ξ) ∀ϕ ∈ W01,∞ Ω, Rm . (2.5) 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 liminf k→∞ Ω f x,Duk dx ≥ Ω f (x,Du)dx, (2.6) for all u ∈ W 1,p (Ω, Rm ), with f (x,Du) ∈ L1 (Ω), and for all uk ∈ W 1,q (Ω, Rm ) such that uk converges to u in the weak topology of W 1,p (Ω, Rm ). Proof. Let u ∈ W 1,p (Ω, Rm ) and uk u in W 1,p . Without loss of generality we may assume that liminf k→∞ Ω f x,Duk dx < ∞. (2.7) Let Ω0 be an open set compactly contained in Ω. For every integer ν such that λ(1/ν) < 1, we consider a subdivision of Ω0 in open sets Ωi such that Ωi ∩ Ω j = ∅ for i = j, i Ωi  = Ω0 , and diamΩi < 1/ν. By (A2) and Remark 2.1, there exists xi ∈ Ωi for which 1 1 + f (x,ξ) , ν 1 f xi ,ξ − f (x,ξ) ≤ µ 1 + f xi ,ξ , ν f xi ,ξ − f (x,ξ) ≤ λ (2.8) I. Benedetti and E. Mascolo 31 for all x ∈ Ωi and ξ ∈ Rnm . Then by using the second inequality of (2.8) Ω0 f (x,Du)dx ≤ Ωi i ≤ Ωi i f (x,Du)dx 1 1 + f xi ,Du dx. ν i Ωi f xi ,Du dx + µ (2.9) We estimate the first integral in the righthand 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 f xi ,Du dx ≤ liminf k→∞ f xi ,Duk dx. Ωi (2.10) The first inequality of (2.8) implies that liminf k→∞ i Ωi f xi ,Duk dx ≤ liminf k→∞ i 1 ≤ 1+λ ν Ωi 1 ν 1+λ liminf k→∞ Ω0 1 ν f x,Duk + λ 1 f x,Duk dx + λ ν dx (2.11) Ω0 . By (2.7) and (2.10), there exists M > 0 such that Ωi i f xi ,Du dx < M (2.12) and consequently we obtain Ω0 f (x,Du)dx 1 ≤ 1+λ liminf ν k→∞ Ω0 f x,Duk dx + λ 1 ν Ω0 + µ 1 ν (2.13) M + Ω0 . Thus we go to the limit as ν → ∞ and then we get the result as Ω0 ↑ 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, Q f (x,ξ) = sup g ≤ f : g quasiconvex with respect to ξ . (2.14) By the results contained in Dacorogna [4] and in Giusti [10], we have that (A1) implies Q f (x,ξ) = inf 1 Ω Ω 1,q f x,ξ + Dϕ(y) d y ∀ϕ ∈ W0 . (2.15) By the definition of Q f , we have that 0 ≤ Q f (x,ξ) ≤ L 1 + ξ q . Now we show that when f satisfies (A2), Q f satisfies the same property. (2.16) 32 Regularity of minimizers for nonconvex integrals Lemma 2.3. Let f : Ω × Rnm → R satisfy (A1) and (A2); then there exists δ > 0 such that for every compact subset Ω0 of Ω with diam(Ω0 ) < δ, there exists x0 ∈ Ω0 such that Q f x0 ,ξ − Q f (x,ξ) ≤ µ x − x0 1 + Q f (x,ξ) , (2.17) for all x ∈ Ω0 and for all ξ ∈ Rnm , where µ(t) = λ(t)/(1 − λ(t)), for 0 < t < δ. Proof. By the characterization of the quasiconvex envelope (2.15), we have that, for x ∈ 1,q Ω, there exists a sequence (ϕx j ) ∈ W0 (Ω, Rm ) such that 1 Ω Q f (x,ξ) ≤ Ω 1 f x,ξ + Dϕx j (y) d y ≤ Q f (x,ξ) + . j (2.18) Let Ω0 Ω be a compact subset and let x0 ∈ Ω0 satisfy (2.2): Q f x0 ,ξ − Q f (x,ξ) 1 1 1 ≤ f x0 ,ξ + Dϕx j (y) d y − f x,ξ + Dϕx j (y) d y + Ω Ω Ω Ω j 1 ≤ λ x − x0 1 + Q f (x,ξ) + λ x − x0 + 1 ; j (2.19) as j → ∞, we obtain Q f x0 ,ξ − Q f (x,ξ) ≤ λ x − x0 1 + Q f (x,ξ) . (2.20) 1,q Now let (ϕx0 j ) ∈ W0 (Ω, Rm ) satisfy (2.18) with x0 in place of x. With the same previous arguments, by applying (2.3) instead of (2.2), for Ω0 with diam(Ω0 ) < δ (with δ chosen such that λ(δ) < 1), we get Q f (x,ξ) − Q f x0 ,ξ ≤ µ x − x0 1 + Q f x0 ,ξ , (2.21) so (2.20) and (2.21) imply (2.17). Remark 2.4. Let f (x,ξ) = a(x)g(ξ), where g : Rnm → R has qgrowth: 0 < g(ξ) < L 1 + ξ q , (2.22) where L > 0 is a constant and a ∈ C 0,α (Ω). In this case, both f and Q f satisfy (A1) and (A2); in fact Q f (x,ξ) = a(x)Qg(ξ). We introduce the functional Ᏺ p,q : Ᏺ p,q (u) = inf liminf uk k→∞ Ω f x,Duk dx, uk ∈ W 1,q Ω, Rm , uk u in W 1,p Ω, Rm The following theorem holds. (2.23) . I. Benedetti and E. Mascolo 33 Theorem 2.5. Let 1 < p < q < pn/(n − 1), let f satisfy (A1) and (A2), and Q f (x,ξ) ∈ L1 (Ω). Then for u ∈ W 1,p (Ω, Rm ), Ᏺ p,q ≥ Ω Q f (x,Du)dx. (2.24) When u ∈ W 1,q (Ω, Rm ), the equality in (2.24) 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 Ω0 in open set Ωi , with diam(Ωi ) < 1/ν, where ν is suﬃciently big, so that Ωi satisfies the condition of Lemma 2.3. Then if (uk ) ∈ W 1,q (Ω, Rm ) and uk u in W 1,p (Ω, Rm ), we obtain liminf k→∞ Ω f x,Duk dx ≥ liminf k→∞ Ω Q f x,Duk dx ≥ Ω Q f (x,Du)dx; (2.25) taking the infimum over all such sequences, (2.24) follows. When u ∈ W 1,q (Ω, Rm ), by the standard relaxation results, there exists a sequence (w j ) ∈ W 1,q (Ω, Rm ) such that w j u in W 1,q (Ω, Rm ) and liminf j →∞ Ω f x,Du j dx = Q f (x,Du)dx, Ω (2.26) which easily implies that (2.24) is an equality. Assume now a pcoercivity condition on f . (A1 ) There exist c0 ,c1 ,L > 0 such that c1 ξ  p − c0 ≤ f (x,ξ) ≤ L 1 + ξ q . (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 Q f (x,ξ) is a convex function with respect to ξ, that is, Q f (x,ξ) = f ∗∗ (x,ξ). Then, for all u ∈ W 1,p (Ω, Rm ) such that Q f (x,Du) ∈ L1loc (Ω), Ᏺ p,q (u,U) = U Q f (x,Du)dx, (2.28) for all U Ω, open set. Proof. Consider a smooth kernel ϕ ≥ 0 in Rm with support in B(0,1), Rn ϕ(x)dx = 1, and ϕ j (x) = j n ϕ( jx). For each j ∈ N, consider ϕ j ∗ u ∈ W 1,q (U, Rm ). Again for the standard relaxation results, we can select a sequence v jk ∈ W 1,q (U, Rm ) such that limk→∞ v jk = ϕ j ∗ u weakly in W 1,q (U, Rm ) and lim k→∞ U f x,Dv jk dx = U Q f x,D ϕ j ∗ u dx. (2.29) 34 Regularity of minimizers for nonconvex integrals We may extract a diagonal subsequence u j = v jk( j) such that u j u in W 1,p (U, Rm ) and f x,Du j dx − U U 1 Q f x,D ϕ j ∗ u dx ≤ . j (2.30) < 1}, we consider a subdivision of U For every positive ν ∈ N with ν ≥ ν = inf {ν : λ(1/ν) in the open sets Ui such that i Ui  = U , Ui U j = ∅, for i = j and diam(Ui ) < 1/ν ∀i. We have, from (2.21), liminf j →∞ U Q f x,D ϕ j ∗ u dx = liminf j →∞ ≤ liminf i Ui i 1 µ ν j →∞ + liminf Ui 1 + Q f xi ,D ϕ j ∗ u Q f xi ,D ϕ j ∗ u Ui i j →∞ Q f x,D ϕ j ∗ u dx 1 1 U  + 1 + µ =µ ν ν liminf dx (2.31) j →∞ Ui i Q f xi ,D ϕ j ∗ u dx. j Since Q f is convex and the measure µx , given by j µx ,v := U ϕ j (x − y)v(y)d y, (2.32) is a probability measure, using Jensen inequality, we have, for fixed i, Ui Q f xi ,D ϕ j ∗ u = Q f xi , Ui j µx ,Du dx ≤ Ui j µx ,Q f xi ,Du dx. (2.33) Since Q f (xi ,Du) ∈ L1 (Ω), by a known result, we get Ui j µx ,Q f xi ,Du dx ≤ Ui Q f xi ,Du dx, (2.34) then, by (2.20), liminf j →∞ U Q f x,D ϕ j ∗ u 1 1 ≤µ U  + 1 + µ ν ν 1 1 ≤µ U  + 1 + µ ν ν + 1+µ 1 ν U i λ Ui Q f xi ,Du dx 1 1 + Q f (x,Du) dx ν i Ui Q f (x,Du)dx. (2.35) I. Benedetti and E. Mascolo 35 As ν goes to ∞, since < ∞, we get U Q f (x,Du)dx liminf j →∞ U Q f x,D ϕ j ∗ u dx ≤ U Q f (x,Du)dx. (2.36) Then, by (2.30), Ᏺ p,q (u,U) ≤ liminf j →∞ U f x,Du j dx = liminf Q f x,D ϕ j ∗ u dx j →∞ U ≤ Q f (x,Du)dx, (2.37) U so by (2.24), (2.28) holds. Remark 2.7. Under the hypotheses of Theorem 2.6, consider the following functional: Ᏺ(u,Ω) = inf liminf uk k→∞ Ω f x,Duk dx, uk ∈ W 1,p Ω, Rm , (2.38) uk u in W 1,p Ω, Rm , f x,Duk ∈ L1 (Ω) . For all U Ω, we have Ᏺ p,q (u,U) = Ᏺ(u,U). (2.39) Indeed for all uk ∈ W 1,p (U, Rm ) with f (x,Duk ) ∈ L1 (U) and uk u in W 1,p (Ω, Rm ), we have Ω f ∗∗ (x,Du)dx ≤ liminf k→∞ Ω f ∗∗ x,Duk dx ≤ liminf k→∞ Ω f x,Duk dx. (2.40) Therefore, Ᏺ(u,U) ≤ Ᏺ (u,U) = p,q U f ∗∗ (x,Du) ≤ Ᏺ(u,U). (2.41) Now we exhibit a class of integrands f : Ω × Rnm → R such that Q f (x,A) = f ∗∗ (x,A), for every A ∈ Rnm . We say that f : Ω × Rnm → R is rankone convex if, for every x ∈ Ω, f x,λA + (1 − λ)B ≤ λ f (x,A) + (1 − λ) f (x,B), (2.42) for all λ ∈ [0,1] and A,B ∈ Rnm with rank(A − B) ≤ 1. For f : Ω × Rnm → R, we define the rankone convex envelope as R f (x,A) = sup{g ≤ f : g rankone convex with respect to A}. We prove the following result. (2.43) 36 Regularity of minimizers for nonconvex integrals Proposition 2.8. Let Ω be a bounded open set of Rn and let f : Ω × Rnm → R. Assume that there exists a function g : Ω × R+ → R such that f (x,A) = g x, A , (2.44) g(x,t) = g(x, −t), for every x ∈ Ω and for every t > 0. Moreover, assume that there exists a measurable, nonnegative function α : Ω → R+ such that g ∗∗ (x,t) = g x,α(x) , (2.45) g ∗∗ (x,t) = g(x,t), (2.46) for all t with t  < α(x) and for all t ≥ α(x). Then, g ∗∗ x, A = f ∗∗ x, A = Q f x, A = R f x, A . (2.47) 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 ε + g ∗∗ x, A ≥ λg(x,b) + (1 − λ)g(x,c), (2.48) A = λb + (1 − λ)c. Choosing B = bA/ A and C = cA/ A, we get ε + g ∗∗ x, A ≥ λ f (x,B) + (1 − λ) f (x,C) ≥ f ∗∗ x, A . (2.49) 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 g ∗∗ (x,v) ≥ g ∗∗ x,α(x) , (2.50) for all v ≥ α(x). In fact, if v > α(x) is such that g ∗∗ (x,v) < g ∗∗ (x,α(x)), then we can choose w < α(x) such that λw + (1 − λ)v = α(x). Since g ∗∗ (x,w) = g ∗∗ (x,α(x)), we get λg ∗∗ (x,w) + (1 − λ)g ∗∗ (x,v) < g ∗∗ x,α(x) = g ∗∗ x,λw + (1 − λ)v (2.51) 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 g ∗∗ (x,v) = g ∗∗ x,α(x) = g ∗∗ (x,w); (2.52) I. Benedetti and E. Mascolo 37 if 0 < w < α(x) < v, we have g ∗∗ (x,v) ≥ g ∗∗ x,α(x) = g ∗∗ (x,w); (2.53) 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 λg ∗∗ (x,v) + (1 − λ)g ∗∗ x,α(x) < λg ∗∗ (x,w) + (1 − λ)g ∗∗ (x,w) = g ∗∗ (x,w) = g ∗∗ x,λv + (1 − λ)α(x) , (2.54) again in contradiction with the convexity of the function g ∗∗ . Since g ∗∗ (x,t) is not decreasing, g ∗∗ (x, A) is a convex function less than f , so we obtain g ∗∗ (x, A) < f ∗∗ (x,A). Therefore, we have proved that g ∗∗ (x, A) = f ∗∗ (x,A). We are going now to prove that R f = f ∗∗ . We observe that g ∗∗ x, A = f ∗∗ (x,A) ≤ R f (x,A). (2.55) When A ≥ α(x), by (2.46), g ∗∗ x, A = g x, A = f (x,A) ≥ R f (x,A). (2.56) So we can reduce to the case A < α(x) when g ∗∗ (x, A) = g ∗∗ (x,α(x)). Let A = (Aij )1≤i≤m, 1≤ j ≤n , with A11 = 0, and define A11 1 λ= 1+ 2 1/2 , 2 α(x)2 − A2 + A11 (2.57) we have 1/2 < λ < 1. Let E = (Eij )1≤i≤m, 1≤ j ≤n be such that E11 = 2A11 , 1 − 2λ Eij = 0, i > 1, j > 1 (2.58) and let B = A − (1 − λ)E, C = A + λE. (2.59) Therefore, we have that A = λB + (1 − λ)C, B  = C  = α(x), rank(B − C) ≤ 1, (2.60) so by the characterization of the rankone convex envelope, R f (x,A) ≤ λ f (x,B) + (1 − λ) f (x,C) = g ∗∗ x,α(x) = f ∗∗ (x,A). (2.61) 38 Regularity of minimizers for nonconvex integrals 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 inf u∈X f (x,Du)dx < inf u∈Y f (x,Du)dx. (2.62) The following result holds. Corollary 2.9. Under the hypotheses of Theorem 2.6, the Lavrentiev phenomenon (2.62) does not occur. Proof. Define, as in Buttazzo and Mizel [1], the Lavrentiev gap: ᏸ(u) = ᏲY (u) − ᏲX (u), ᏸ(u) = 0 if ᏲX (u) = +∞, (2.63) where ᏲX (u) = sup G : X −→ [0,+∞] : G l.s.c, G ≤ I on X , (2.64) ᏲY (u) = sup G : X −→ [0,+∞] : G l.s.c, G ≤ I on Y . Since in our case X = W 1,p (U, Rm ) and Y = W 1,q (U, Rm ), U Ω and ᏲY (u,U) = Ᏺ p,q (u,U) = I ∗∗ (u,U) ≤ ᏲX (u,U), (2.65) we conclude that ᏸ(u) = 0. Therefore, Proposition 2.8 exhibits a class of functions satisfying the condition (B1) (i.e., f (x,ξ) = g(x, ξ )) for which the Lavrentiev phenomenon does not occur. 3. 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 × Rnm → 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 ξ ∈ Rnm \ BR (0), f (x,ξ) = g x, ξ  ; (3.1) (B2) f is puniformly 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 ξ1 ,ξ2 ∈ Rnm \ BR (0) endpoints of a segment contained in the complement of BR (0), 1 ξ +ξ f x,ξ1 + f x,ξ2 ≥ f x, 1 2 + ν 1 + ξ1 2 2 2 + ξ2 2 (p−2)/2 2 ξ1 − ξ2 ; (3.2) I. Benedetti and E. Mascolo 39 (B3) for almost everywhere x ∈ Ω and every ξ ∈ Rnm \ BR (0), let Dt+ g(x, ξ ) be the rightside derivative of g with respect to t and denote ξiα Dξ+iα f (x,ξ) = Dt+ g x, ξ  ξ  . (3.3) Then for every ξ ∈ Rnm \ BR (0), the vector field x → Dξ+ f (x,ξ) is weakly diﬀerentiable and q −1 Dx Dξ+ f (x,ξ) ≤ L1 1 + ξ  Let . (3.4) I(v,Ω) = Ω f (x,Dv)dx. (3.5) 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. Theorem 3.1. Let u be a local minimizer of (3.5) whose integrand f is convex with respect to ξ and satisfies the assumptions (A1), (B1), (B2), and (B3), 1 < p ≤ q < p(n + 1)/n. Then u is locally Lipschitz continuous, and, for all Br (x0 ) Ω, sup Du ≤ c Br/4 (x0 ) Br (x0 ) β 1 + f (x,Du) dx , (3.6) where c = c(n, p, q,L,L1 ,R,ν) and β = β(n, p, q). In the sequel, we need some properties of functions satisfying (A1), (A2), and the puniformly convexity at infinity. In [2, Lemma 2.2], it is proved that assumption (B2) implies that f is pcoercive, that is, there exist c0 ,c1 > 0 such that c1 ξ  p − c0 ≤ f (x,ξ). (3.7) By [2, Theorem 2.5(iv)], it follows that if f satisfies the condition (B2), there exists R0 depending on ν, p, q, L such that f ∗∗ (x,ξ) = f (x,ξ), (3.8) for almost everywhere x ∈ Ω and ξ ∈ Rnm \ BR0 (0). Moreover, by assumption (A2) and Remark 2.1, it is easy to check that there exists δ > 0 such that, for every ξ ∈ BR0 (0) and x ∈ U, with U Ω and diamU < δ, there exists x0 ∈ U such that f (x,ξ) ≤ max f x0 ,ξ 1 + µ(δ) + µ(δ), ξ ∈BR0 (0) (3.9) then, for every ξ1 ,ξ2 ∈ BR0 (0), f x,ξ1 − f x,ξ2 where c, c2 are positive constants. ≤ c max f x0 ,ξ + 1 = c2 R0 ,U , ξ ∈B R 0 (3.10) 40 Regularity of minimizers for nonconvex integrals We now consider ξ1 ,ξ2 ∈ Rnm \ BR0 (0). By (3.8) and the growth condition, we obtain f x,ξ1 − f x,ξ2 ≤ c3 1 + ξ1 + ξ2 q −1 ξ1 − ξ2 , (3.11) with c3 = c3 (L, q). Therefore, it is easy to check that there exist c2 = c2 (R0 ,U) and c3 (L, q) such that f x,ξ1 − f x,ξ2 q −1 ≤ c2 + c3 1 + ξ1 + ξ2 ξ1 − ξ2 . (3.12) Finally, for every x ∈ U, where U Ω, and for every ξ ∈ Rnm , (3.8) implies that f (x,tξ) ≤ c4 1 + f (x,ξ) , (3.13) for all t ∈ [0,1], where c4 depends on ν and R0 . Our main result is the following regularity theorem. Theorem 3.2. Let f satisfy (A1), (A2), (B1), (B2), and (B3), with 1 < p < q < p(n + 1)/n and Q f (x,ξ) = f ∗∗ (x,ξ). Let u be a local minimizer of the functional I in (3.5), then u ∈ 1,∞ Wloc (Ω, Rm ) and estimate (3.6) holds. Proof. We suppose that for every U Ω, u is a local minimizer and then u is a solution of the following boundary value problem: 1,p Ω, Rm + u . I(u) = inf I(v,U), v ∈ W0 (3.14) Consider the relaxed problem of (3.14): 1,p I ∗∗ (u) = inf I ∗∗ (v,U), v ∈ W0 Ω, Rm + u , (3.15) where, as usual, I ∗∗ (v,U) = U f ∗∗ (x,Dv)dx. (3.16) 1,p By the convexity of I ∗∗ , the problem (3.15) has at least a solution u ∈ W0 (Ω, Rm ) + u. Since u is also a local minimizer of I ∗∗ and f ∗∗ satisfies (A1), (B1), (B2), and (B3) in 1,∞ (U, Rm ). Ω × Rnm \ BR0 (0), then we can apply Theorem 3.1 and so we obtain u ∈ Wloc Observe that f (x,Du) ∈ L1 (U). In fact, we can suppose, without loss of generality, that diamU < δ; then, for ξ ∈ BR0 (0), by (3.9), f (x,ξ) ≤ c2 R0 ,U ; (3.17) and then we get U f (x,Du)dx = U ∩{x:Du≤R0 } f (x,Du)dx + ≤ c2 R0 ,U U  + U U ∩{x:Du>R0 } f ∗∗ (x,Du)dx < ∞. f ∗∗ (x,Du)dx (3.18) I. Benedetti and E. Mascolo 41 Let (uk ) ⊂ W 1,q (U, Rm ), with uk u w − W 1,p (U, Rm ); for ε > 0, let Σ U, Σ open, such that I(u,U \ Σ) ≤ ε; (3.19) Du∞,Σ ≤ M. (3.20) and let M > 0 be such that Moreover, let Σ0 Σ and let ν ∈ N. For i = 1,...,ν, we define Σi = x ∈ Σ : dist x,Σ0 i < R , ν (3.21) where R = dist(Σ0 ,∂Σ). For i = 1,...,ν, consider the scalar functions ϕi ∈ C01 (Σi ) such that 0 ≤ ϕi ≤ 1, 1 in Σi−1 , ϕi (x) = 0 in Σ \ Σi , Dϕi ≤ ν+1 . R (3.22) For all k ∈ N, define vki = 1 − ϕi u + ϕi uk . (3.23) We have vki = uk in Σi−1 and vki = u in Σ \ Σi−1 . Define ṽki = vki in Σ and ṽki = u in U \ Σ. Consider f x,Dṽki dx = U U \Σ + then we obtain f (x,Du)dx + U Σi−1 Σ\Σi f x,Duk dx + + Σ\Σi f (x,Du)dx + Σi \Σi−1 + Σi \Σi−1 f x,Dṽki dx ≤ ε + f (x,Du)dx Σi \Σi−1 Σi−1 f x,Duk dx f x,Dvki − f x,ϕi Duk dx (3.24) f x,Dvki dx, (3.25) f x,ϕi Duk dx. Inequality (3.12) gives Jki = Σi \Σi−1 f x,Dvki − f x,ϕi Duk dx ≤ c2 Σi \ Σi−1 + c3 ≤ c2 Σi \ Σi−1 + c3 Σi \Σi−1 Σi \Σi−1 1 + Dvki + ϕi Duk q −1 1 + Du + 2 Duk + Dϕi × Dvki − ϕi Duk dx uk − u 1 − ϕi Du + Dϕi q −1 uk − u dx. (3.26) 42 Regularity of minimizers for nonconvex integrals By applying the Holder inequality with the exponents p/(q − 1) and p/(p − q + 1), we obtain q −1 ν+1 Jki ≤ c2 Σ \ Σ0 + c3 1 +  Du  + 2 Du + u − u k k p R L i=1 ν (p−q+1)/ p × M Σ \ Σ0 ν+1 uk − u p/(p−q+1) + L (Σν ) . R (3.27) Taking in account that the sequence (Duk ) has a bounded norm in L p (Ω, Rnm ) and that, since the assumption q < p(n + 1)/n implies p/(p − q + 1) < p∗ and (uk ) converges s − L p/(p−q+1) (Ω, Rm ) to u, we can conclude that there exists a constant c5 independent of k and ν such that limsup k→∞ ν i=1 Jki ≤ c5 . (3.28) 1,p Observe that, since vki ∈ W0 (U, Rm ) + u and u = u on ∂U, by the minimality of u for problem (3.14), we get I(u,U) ≤ I(ṽki ,U), for all i for all k. Therefore, summing up with respect to i = 1,...,ν in (3.25), we get ν U f (x,Du)dx ≤ νε + ν f (x,Du)dx + ν i=1 Σ\Σi ν + c5 + i=1 Σi \Σi−1 Σ f x,Duk dx (3.29) f x,ϕi Duk dx. Moreover, since (3.13) implies that f x,ϕi Duk ≤ c4 1 + f x,Duk , (3.30) we obtain U f (x,Du)dx ≤ ε + ν + c6 ν U f x,Duk dx + c5 1 + ν ν Σ\Σ0 f (x,Du)dx. (3.31) As k goes to infinity, we get U ν + c6 f (x,Du)dx ≤ ε + liminf ν k→∞ U c 1 f x,Duk dx + 5 + ν ν Σ\Σ0 f (x,Du)dx. (3.32) Then passing to the limit ν → ∞, U f (x,Du)dx ≤ ε + liminf k→∞ U f x,Duk dx. (3.33) Taking the infimum over the sequences (uk ), U f (x,Du)dx ≤ Ᏺ p,q (u,U). (3.34) I. Benedetti and E. Mascolo 43 Since Q f (x,ξ) = f ∗∗ (x,ξ), by Theorem 2.6, we have f (x,Du)dx ≤ U U f ∗∗ (x,Du)dx. (3.35) On the other hand, u is a solution of (3.15), then U f (x,Du)dx = U f ∗∗ (x,Du)dx = U f ∗∗ (x,Du)dx, (3.36) 1,∞ which implies that u is also a solution of (3.15). Theorem 2.2 ensures that u ∈ Wloc (U, 1, ∞ m m R ); for the arbitrariness of U, we get u ∈ Wloc (Ω, R ). Estimate (3.6) follows easily. Remark 3.3. We observe that the arguments of the proof of Theorem 3.2 show that, for 1,∞ (Ω, Rm ) and for all U Ω, we have all u ∈ Wloc Ᏺ0 (u,U) = Ᏺ p,q (u,U) = U f ∗∗ (x,Du)dx, (3.37) where Ᏺ0 (u,U) = inf liminf uk k→∞ Ω 1,p Ω, Rm + u, f x,Duk dx, uk ∈ W0 (3.38) uk u in W 1,p Ω, Rm , f x,Duk ∈ L1 (Ω) . 1,p Indeed, back to the proof of Theorem 3.2, the sequence ṽki ∈ W0 (U, Rm ) + u is such that f (x,Dṽki ) ∈ L1 (U) and ṽki u in W 1,p (Ω, Rm ). Therefore, taking the lower bound in (3.25) and summing up with respect to i = 1,...,ν as k → ∞, we have Ᏺ0 (u,U) ≤ ε + ν + c6 liminf ν k→∞ U f x,Duk dx + c5 1 + ν ν Σ\Σ0 f (x,Du)dx. (3.39) As ν → ∞, we get Ᏺ0 (u,U) ≤ Ᏺ p,q (u,U) ≤ Ᏺ0 (u,U). (3.40) References [1] [2] [3] [4] [5] G. Buttazzo and V. J. Mizel, Interpretation of the Lavrentiev phenomenon by relaxation, J. Funct. Anal. 110 (1992), no. 2, 434–460. G. Cupini, M. Guidorzi, and E. Mascolo, Regularity of minimizers of vectorial integrals with pq growth, Nonlinear Anal. 54 (2003), no. 4, 591–616. G. 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Fusco, On the convergence of integral functionals depending on vectorvalued functions, Ricerche Mat. 32 (1983), no. 2, 321–339. E. Giusti, Metodi Diretti nel Calcolo delle Variazioni [Direct Methods in the Calculus of Variations], Unione Matematica Italiana, Bologna, 1994 (Italian). P. Marcellini, Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals, Manuscripta Math. 51 (1985), no. 1–3, 1–28. , Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Ration. Mech. Anal. 105 (1989), no. 3, 267–284. E. Mascolo and A. P. Migliorini, Everywhere regularity for vectorial functionals with general growth, ESAIM Control Optim. Calc. Var. 9 (2003), 399–418. E. Mascolo and R. Schianchi, Existence theorems in the calculus of variations, J. Diﬀerential Equations 67 (1987), no. 2, 185–198. Irene Benedetti: Dipartimento di Matematica “Ulisse Dini,” Università di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy Email address: benedetti@math.unifi.it Elvira Mascolo: Dipartimento di Matematica “Ulisse Dini,” Università di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy Email address: mascolo@math.unifi.it Advances in Operations Research Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Advances in Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Applied Mathematics Algebra Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Probability and Statistics Volume 2014 The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Differential Equations Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014 Submit your manuscripts at http://www.hindawi.com International Journal of Advances in Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Mathematics and Mathematical Sciences Mathematical Problems in Engineering Journal of Mathematics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Discrete Mathematics Journal of Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Discrete Dynamics in Nature and Society Journal of Function Spaces Hindawi Publishing Corporation http://www.hindawi.com Abstract and Applied Analysis Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Journal of Stochastic Analysis Optimization Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014