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

TL;DR

This paper studies the stability of solutions to obstacle-type variational inequalities under Mosco-convergence of constraint sets, for differential operators with lower order terms of natural growth. It extends classical stability results to a Leray–Lions-type principal part with a lower-order term $H(x,u,Du)$ that grows like $|Du|^p$, establishing that, when obstacle sets converge in the Mosco sense and obstacles are bounded in $L^{\infty}$, the solutions converge (up to subsequences) in $W_0^{1,p}(\Omega)$ to a solution of the limiting obstacle problem. The analysis builds on a priori estimates and approximation schemes (via truncated growth $H_j$) inspired by Boccardo–Murat–Puel, ensuring uniform $L^{\infty}$ and $W^{1,p}$ bounds and passing to the limit. The results generalize classical obstacle problem stability to natural-growth settings, broadening applicability in variational inequalities constrained by convex obstacle sets. This contributes to the understanding of how solution behavior persists under domain and constraint perturbations in nonlinear PDEs with lower-order nonlinearities.

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 $ψ\in W_{0}^{1,p}(Ω)\cap L^{\infty}(Ω)$, we consider $u\in W_{0}^{1,p}(Ω)\cap L^{\infty}(Ω)$ satisfying $u\geqψ$ a.e. and $$ \langle A(u),v-u\rangle+\int_ΩH(x,u,\nabla u)(v-u)\geq 0$$ for all $v\in W_{0}^{1,p}(Ω)\cap L^{\infty}(Ω)$ with $v\geqψ$. Here, $A$ is a Leray-Lions type operator, mapping $W_0^{1,p}(Ω)$ into its dual $W^{-1, p'}(Ω)$, while $H(x, u, D u)$ grows like $|D u|^p$. 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.

Theorems & Definitions (7)

• proof
• proof : Proof of Lemma \ref{['LEMMA:disug:stimainfty']}
• proof
• proof
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• proof
• proof : Proof of \ref{['Theoprincipale']}