Mosco-convergence of convex sets and unilateral problems for differential operators with lower order terms having natural growth

Abstract

We study the stability of solutions to a class of variational inequalities posed on obstacle-type convex sets, under Mosco-convergence. More precisely, for a fixed obstacle $\psi \in W_0^{1,p}(\Omega )\cap L^{\infty }(\Omega )$, we consider $u\in W_0^{1,p}(\Omega )\cap L^{\infty }(\Omega )$ satisfying $u\ge \psi$ a.e. and $$⟨A(u),v-u⟩+{\int }_{\Omega }H(x,u,\nabla u)(v-u)\ge 0$$ for all $v\in W_0^{1,p}(\Omega )\cap L^{\infty }(\Omega )$ with $v\ge \psi$. Here, $A$ is a Leray-Lions type operator, mapping $W_0^{1,p}(\Omega )$ into its dual $W^{-1,p^{\prime }}(\Omega )$, while $H(x,u,Du)$ grows like $|Du{|}^p$. Our main result establishes that the solutions are stable under Mosco-convergence of the constraint sets. This extends classical stability results to natural growth problems.