Uniform convexity and variational convergence
Vasilii Vasil'evich Zhikov, Svetlana Evgenievna Pastukhova · Transactions of the Moscow Mathematical Society · 2014
Let Ω be a domain in R d .We establish the uniform convexity of the Γ-limit of a sequence of Carathéodory integrands f (x, ξ) : Ω×R d → 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.