Uniform convexity and variational convergence

V. V. Zhikov ; S. E. Pastukhova

Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 75 (2014) no. 2, pp. 245-276

Résumé

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 $ \vert\xi \vert^{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.