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

Published online by Cambridge University Press:  15 September 2002

Alessandra Coscia  and
Domenico Mucci
Alessandra Coscia
Affiliation:
Dipartimento di Matematica, Università di Parma, via M. D'Azeglio 85/A, 43100 Parma, Italy; [email protected].
Domenico Mucci
Affiliation:
Dipartimento di Matematica, Università di Parma, via M. D'Azeglio 85/A, 43100 Parma, Italy; [email protected].
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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.

Keywords

Integral representationΓ-convergencenonstandard growth conditions.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 7 , 2002 , pp. 495 - 519
Copyright
© EDP Sciences, SMAI, 2002

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