$\Gamma$-convergence for power-law functionals with variable exponents

Abstract

We study the $\Gamma$-convergence of the functionals $F_n(u):=||f(\cdot ,u(\cdot ),Du(\cdot ))|{|}_{p_n(\cdot )}$ and ${\mathcal{F}}_n(u):={\int }_{\Omega }\frac{1}{p_n(x)}{f}^{p_n(x)}(x,u(x),Du(x))dx$ defined on $X\in \{L^1(\Omega ,{\mathbb{R}}^d),L^{\infty }(\Omega ,{\mathbb{R}}^d),C(\Omega ,{\mathbb{R}}^d)\}$ (endowed with their usual norms) with effective domain the Sobolev space $W^{1,p_n(\cdot )}(\Omega ,{\mathbb{R}}^d)$. Here $\Omega \subseteq {\mathbb{R}}^N$ is a bounded open set, $N,d\ge 1$ and the measurable functions $p_n:\overline{\Omega }\to (1,+\infty )$ satisfy the conditions ${\mathrm{esssup}}_{\overline{\Omega }}{p}_n\le \beta {\mathrm{essinf}}_{\overline{\Omega }}{p}_n$ for a fixed constant $\beta >1$ and ${\mathrm{essinf}}_{\overline{\Omega }}{p}_n\to +\infty$ as $n\to +\infty$. We show that when $f(x,u,\cdot )$ is level convex and lower semicontinuous and it satisfies a uniform growth condition from below, then, as $n\to \infty$, the sequences $(F_n{)}_n$ $\Gamma$-converges in $X$ to the functional $F$ represented as $F(u)=||f(\cdot ,u(\cdot ),Du(\cdot ))|{|}_{\infty }$ on the effective domain $W^{1,\infty }(\Omega ,{\mathbb{R}}^d)$. Moreover we show that the $\Gamma$-${\lim }_n{\mathcal{F}}_n$ is given by the functional $\mathcal{F}(u):=\{\begin{array}{lll}& 0& if\; ||f(\cdot ,u(\cdot ),Du(\cdot ))|{|}_{\infty }\le 1,\\ & +\infty & otherwise\; inX.\end{array}$