Integral representation and $\Gamma$-convergence of variational integrals with $p\left(x\right)$-growth
ESAIM: Control, Optimisation and Calculus of Variations, Volume 7 (2002), pp. 495-519.

We study the integral representation properties of limits of sequences of integral functionals like $\int f\left(x,Du\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x$ under nonstandard growth conditions of $\left(p,q\right)$-type: namely, we assume that

 ${|z|}^{p\left(x\right)}\le f\left(x,z\right)\le {L\left(1+|z|}^{p\left(x\right)}\right)\phantom{\rule{0.166667em}{0ex}}.$
Under weak assumptions on the continuous function $p\left(x\right)$, we prove $\Gamma$-convergence to integral functionals of the same type. We also analyse the case of integrands $f\left(x,u,Du\right)$ depending explicitly on $u$; finally we weaken the assumption allowing $p\left(x\right)$ to be discontinuous on nice sets.

DOI: 10.1051/cocv:2002065
Classification: 49J45,  49M20,  46E35
Keywords: integral representation, $\Gamma$-convergence, nonstandard growth conditions
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author = {Coscia, Alessandra and Mucci, Domenico},
title = {Integral representation and $\sf \Gamma$-convergence of variational integrals with ${p(x)}$-growth},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {495--519},
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Coscia, Alessandra; Mucci, Domenico. Integral representation and $\sf \Gamma$-convergence of variational integrals with ${p(x)}$-growth. ESAIM: Control, Optimisation and Calculus of Variations, Volume 7 (2002), pp. 495-519. doi : 10.1051/cocv:2002065. http://www.numdam.org/articles/10.1051/cocv:2002065/

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