Nonlinear elliptic variational inequalities with unilateral pointwise functional constraints in variable domains

A. A. Kovalevsky<sup>ab

a</sup> N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
^b Institute of Natural Sciences and Mathematics, Ural Federal University

Abstract: We consider variational inequalities with operators ${\mathcal A}_{s}\colon W^{1,p}(\Omega_{s})\to(W^{1,p}(\Omega_{s}))^{\ast}$ in divergence form and constraint sets $V_{s}=\{v\in W^{1,p}(\Omega_{s}): h(v)+\Phi_{s}(v) \leqslant\varphi_{s} \ \text{a.e. in} \ \Omega_{s}\}$, where $\Omega_s$ with $s\in\mathbb N$ is a domain in $\mathbb R^n$ contained in a bounded domain $\Omega\subset\mathbb R^n$ ($n\geqslant 2$), $p>1$, $h$ is a convex function on $\mathbb R$, $\varphi_{s}$ is a function on $\Omega_{s}$, and $\Phi_{s}$ is a continuous convex functional on $W^{1,p}(\Omega_{s})$. We describe conditions for a weak convergence of solutions of the considered variational inequalities to the solution of a variational inequality with an operator from $W^{1,p}(\Omega)$ to $(W^{1,p}(\Omega))^{\ast}$ and constraint set defined by the equality $V=\{v\in W^{1,p}(\Omega): h(v)+\Phi(v)\leqslant\varphi \ \text{a.e. in} \ \Omega\}$, where $\varphi$ is a limit function for $\varphi_{s}$ and $\Phi$ is a limit functional for $\Phi_{s}$. These conditions include some requirements on the involved domains, operators, and the mappings defining the constraint sets. In so doing, one of the main conditions is the $G$-convergence of the sequence $\{{\mathcal A}_{s}\}$ to an operator ${\mathcal A}\colon W^{1,p}(\Omega)\to(W^{1,p}(\Omega))^{\ast}$.

Keywords: nonlinear elliptic variational inequality, pointwise functional constraint, variable domains, $G$-convergence of operators, convergence of solutions.

MSC: 47J20, 49J40, 49J45

Received: 15.10.2025
Revised: 28.10.2025
Accepted: 03.11.2025

Language: English

DOI: 10.21538/0134-4889-2025-31-4-132-148

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