Uniform convexity and variational convergence

Abstract:

Let $\Omega$ be a domain in $\mathbb {R}^d$. We establish the uniform convexity of the $\Gamma$-limit of a sequence of Carathéodory integrands $f(x,\xi )\colon \Omega { \times }\mathbb {R}^d\to \mathbb {R}$ subjected to a two-sided power-law estimate of coercivity and growth with respect to $\xi$ with exponents $\alpha$ and $\beta$, $1<\alpha \le \beta <\infty$, and having a common modulus of convexity with respect to $\xi$. In particular, the $\Gamma$-limit of a sequence of power-law integrands of the form $|\xi |^{p(x)}$, where the variable exponent $p\colon \Omega \to [\alpha ,\beta ]$ is a measurable function, is uniformly convex.

We prove that one can assign a uniformly convex Orlicz space to the $\Gamma$-limit of a sequence of power-law integrands. A natural $\Gamma$-closed extension of the class of power-law integrands is found.

Applications to the homogenization theory for functionals of the calculus of variations and for monotone operators are given.