On the convergence of solutions of variational problems with variable implicit pointwise constraints in variable domains

Abstract

In this paper, we give sufficient conditions for the convergence of minimizers and minimum values of integral and more general functionals Js: W1 , p(Ωs) → R on the sets Us(hs) = { v∈ W1 , p(Ωs) : hs(v) ⩽ 0 a.e. in Ωs} , where p> 1 , { Ωs} is a sequence of domains contained in a bounded domain Ω of Rn (n⩾ 2), and { hs} is a sequence of functions on R. In so doing, we assume that the considered functionals Γ-converge to a functional defined on W1 , p(Ω) and the spaces W1 , p(Ωs) are strongly connected with the space W1 , p(Ω). Certain conditions on the relation between the functions hs and a function h: R→ R are also required in our main results. © 2018, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature.