Calculus of Variations and Geometric Measure Theory

A. Coscia - D. Mucci

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

created on 26 Jul 2001
modified on 13 Nov 2002


Published Paper

Inserted: 26 jul 2001
Last Updated: 13 nov 2002

Volume: 7
Pages: 495-519
Year: 2002


We study the integral representation properties of limits of sequences of integral functionals like \,$\int f(x,Du)\,dx$\, 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 $\px$, we prove $\Gamma$-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.

Keywords: Gamma-convergence, Integral representation, Nonstandard growth conditions