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Integral representation and $\sf \Gamma $-convergence of variational integrals with ${p(x)}$-growth

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cocv240.dvi ESAIM: Control, Optimisation and Calculus of Variations July 2002, Vol. 7, 495–519 URL: http://www.emath.fr/cocv/ DOI: 10.1051/cocv:2002065 INTEGRAL REPRESENTATION AND Γ-CONVERGENCE OF VARIATIONAL INTEGRALS WITH P (X)-GROWTH Alessandra Coscia1 and Domenico Mucci1 Abstract. We study the integral representation properties of limits of sequences of integral functionals like � f(x,Du) dx under nonstandard growth conditions of (p, q)-type: namely, we assume that |z|p(x) ≤ f(x, z) ≤ L(1 + |z|p(x)) . Under weak assumptions on the continuous function p(x), we prove Γ-convergence to integral func- tionals of the same type. We also analyse the case of integrands f(x, u,Du) depending explicitly on u; finally we weaken the assumption allowing p(x) to be discontinuous on nice sets. Mathematics Subject Classification. 49J45, 49M20, 46E35. Received July 6, 2001. Introduction The aim of this paper is the study of the Γ-convergence and integral representation properties for sequences of integral functionals of the type F (u,Ω) := ∫ Ω f(x, u(x), Du(x)) dx, (0.1) where Ω is an open subset of Rn and f is a non-negative Borel function defined on Ω× RN × RnN . Under the assumption of p-growth |z|p ≤ f(x, u, z) ≤ L(1 + |z|p) (0.2) existence and integral representation of the Γ-limit with respect to the strong topology of Lp of a sequence of functionals as (0.1) was proved in the scalar case in [11, 15, 16], and in the vector-valued case in [21], under suitable assumptions on the dependence of f on u, see also [10, 14]. In the context of regularity theory for minimizers, ten years ago Marcellini [22] replaced (0.2) with the more flexible (p, q)-growth assumption |z|p ≤ f(x, u, z) ≤ L(1 + |z|q) , q ≥ p > 1 ; (0.3) Keywords and phrases: Integral representation, Γ-convergence, nonstandard growth conditions. 1 Dipartimento di Matematica, Universita` di Parma, via M. D’Azeglio 85/A, 43100 Parma, Italy; e-mail: [email protected] c© EDP Sciences, SMAI 2002 496 A

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