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

Coscia, Alessandra; Mucci, Domenico

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 $\int f(x,Du)\mathrm{d}x$ under nonstandard growth conditions of $(p,q)$-type: namely, we assume that

$${\left|z\right|}^{p\left(x\right)}\le f(x,z)\le {L(1+|z|}^{p\left(x\right)}).$$

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 $f(x,u,Du)$ depending explicitly on $u$; finally we weaken the assumption allowing $p\left(x\right)$ to be discontinuous on nice sets.

Classification: 49J45,  49M20,  46E35
Keywords: integral representation, $\Gamma$-convergence, nonstandard growth conditions

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