ESAIM: Control, Optimisation and Calculus of Variations

Research Article

Integral representation and Γ-convergence of variational integrals with p(x)-growth

Coscia, Alessandraa1 and Mucci, Domenicoa1

a1 Dipartimento di Matematica, Università di Parma, via M. D'Azeglio 85/A, 43100 Parma, Italy; domenico.mucci@unipr.it.

Abstract

We study the integral representation properties of limits of sequences of integral functionals like   $\int f(x,Du)\,{\rm d}x$   under nonstandard growth conditions of (p,q)-type: namely, we assume that $$
\vert z\vert^{p(x)}\leq f(x,z)\leq L(1+\vert z\vert^{p(x)})\,.
$$ Under weak assumptions on the continuous function p(x), we prove Γ-convergence to integral functionals 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.

(Received July 6 2001)

(Online publication September 15 2002)

Key Words:

  • Integral representation;
  • Γ-convergence;
  • nonstandard growth conditions.

Mathematics Subject Classification:

  • 49J45;
  • 49M20;
  • 46E35
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