Uniform convexity and variational convergence

Abstract

Let Ω be a domain in Rd. We establish the uniform convexity of the Γ-limit of a sequence of Carathéodory integrands f(x, ξ) : Ω×Rd → R subjected to a two-sided power-law estimate of coercivity and growth with respect to ξ with exponents α and β, 1 < α ≤ β < ∞, and having a common modulus of convexity with respect to ξ. In particular, the Γ-limit of a sequence of power-law integrands of the form |ξ|p(x), where the variable exponent p : Ω → [α, β] is a measurable function, is uniformly convex. We prove that one can assign a uniformly convex Orlicz space to the Γ-limit of a sequence of power-law integrands. A natural Γ-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.