On Integral Representation of Γ-Limit Functionals

Abstract

We consider the Γ-convergence of a sequence of integral functionals F_n(u), defined on the functions u from the Sobolev space W^1,α(Ω) (α > 1); Ω is a bounded Lipschitz domain, where the integrand f_n(x, u,∇u) depends on a function u and its gradient ∇u. As functions of ξ, the integrands f_n(x, s, ξ) are convex and satisfy a two-sided power estimate on the coercivity and growth with different exponents α < β. Moreover, the integrands f_n(x, s, ξ) are equi-continuous over s in some sense with respect to n. We prove that for the functions from L ∞ (Ω) ∩ W^1,β(Ω) the Γ-limit functional coincides with an integral functional F(u) for which the integrand f(x, s, ξ) is of the same class as f_n(x, s, ξ).