Variational Problems with Unilateral Pointwise Functional Constraints in Variable Domains

Abstract

We consider a sequence of convex integral functionals Fs: W1,p(Ωs) → ℝ and a sequence of weakly lower semicontinuous and generally nonintegral functionals Gs: W1,p(Ωs) → ℝ, where {Ωs} is a sequence of domains in ℝn contained in a bounded domain Ω ⊂ ℝn (n ≥ 2) and p > 1. Along with this, we consider a sequence of closed convex sets Vs = {v ∈ W1,p(Ωs): v ≥ Ks(v) a.e. in Ωs}, where Ks is a mapping from the space W1,p(Ωs) to the set of all functions defined on Ωs. We establish conditions under which minimizers and minimum values of the functionals Fs + Gs on the sets Vs converge to a minimizer and the minimum value of a functional on the set V = {v ∈ W1,p(Ω): v ≥ K(v) a.e. in Ω}, where K is a mapping from the space W1,p(Ω) to the set of all functions defined on Ω. These conditions include, in particular, the strong connectedness of the spaces W1,p(Ωs) with the space W1,p(Ω), the condition of exhaustion of the domain Ω by the domains Ωs, the Γ-convergence of the sequence {Fs} to a functional F: W1,p(Ω) → ℝ, and a certain convergence of the sequence {Gs} to a functional G: W1,p(Ω) → ℝ. We also assume some conditions characterizing both the internal properties of the mappings Ks and their relation to the mapping K. In particular, these conditions admit the study of variational problems with irregular varying unilateral obstacles and with varying constraints combining the pointwise dependence and the functional dependence of the integral form.