Formula omitted]-convergence for power-law functionals with variable exponents

Abstract

We study the Γ-convergence of the functionals Fn(u):=||f(⋅,u(⋅),Du(⋅))||pn(⋅) and Fn(u):=∫Ω1pn(x)fpn(x)(x,u(x),Du(x))dx defined on X∈{L1(Ω,Rd),L∞(Ω,Rd),C(Ω,Rd)} (endowed with their usual norms) with effective domain the Sobolev space W1,pn(⋅)(Ω,Rd). Here Ω⊆RN is a bounded open set, N,d≥1 and the measurable functions pn:Ω→[1,+∞) satisfy the conditions ess supΩpn≤βess infΩpn<+∞ for a fixed constant β>1 and ess infΩpn→+∞ as n→+∞. We show that when f(x,u,⋅) is level convex and lower semicontinuous and it satisfies a uniform growth condition from below, then, as n→∞, the sequence (Fn)nΓ-converges in X to the functional F represented as F(u)=||f(⋅,u(⋅),Du(⋅))||∞ on the effective domain W1,∞(Ω,Rd). Moreover we show that the Γ-limnFn is given by the functional F(u):=0if||f(⋅,u(⋅),Du(⋅))||∞≤1,+∞otherwiseinX.