On the convergence of solutions of variational problems with 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, in general, 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) p > 1. Along with this, we consider a sequence of closed convex sets Vs = {v ∈ W1,p(Ωs) : Ms(v) ⩽ 0 a.e. in Ωs}, where Ms is a mapping from 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) : M(v) ⩽ 0 a.e. in Ω}, where M is a mapping from W1,p(Ω) to the set of all functions defined on Ω. © 2021, Springer Science+Business Media, LLC, part of Springer Nature.