This article is cited in
1 paper
MATHEMATICS
Nonlinear variational inequalities with bilateral constraints coinciding on a set of positive measure
A. A. Kovalevskyab a N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
b Ural Federal University, Yekaterinburg, Russian Federation
Abstract:
We consider variational inequalities with invertible operators $\mathcal A_s\colon W^{1,p}_0(\Omega)\to W^{-1,p'}(\Omega)$,
$s\in\mathbb N$, in divergence form and with constraint set $V=\{v\in W^{1,p}_0(\Omega):\varphi\leq v\leq \psi\}$ a.e. in
$\Omega\}$, where
$\Omega$ is a nonempty bounded open set in
$\mathbb R^n$ (
$n\geq2$),
$p>1$, and
$\varphi,\psi\colon\Omega\to\bar{\mathbb R}$ are measurable functions. Under the assumptions that the operators
$\mathcal A_s$ $G$-converge to an invertible operator $\mathcal A\colon W^{1,p}_0(\Omega)\to W^{-1,p'}(\Omega)$, $\operatorname{int}\{\varphi=\psi\}\neq\varnothing$, $\operatorname{meas}(\partial\{\varphi=\psi\}\cap\Omega)=0$, and there exist functions
$\varphi,\overline{\psi}\in W^{1,p}_0(\Omega)$ such that
$\varphi\leq\overline{\varphi}\leq\psi$ a.e. in
$\Omega$ and $\operatorname{meas}(\{\varphi\neq\psi\}\setminus\{\overline{\varphi}\neq\overline{\psi}\})=0$, we establish that the solutions
$u_s$, of the variational inequalities converge weakly in
$W^{1,p}_0(\Omega)$ to the solution
$u$ of a similar variational inequality with the operator
$\mathcal A$ and the constraint set
$V$. The fundamental difference of the considered case from the previously studied one in which
$\operatorname{meas}\{\varphi=\psi\}=0$ is that, in general, the functionals
$\mathcal A_su_s$ do not converge to
$\mathcal Au$ even weakly in
$W^{-1,p'}(\Omega)$ and the energy integrals
$\langle\mathcal A_su_s,u_s\rangle$ do not converge to
$\langle\mathcal Au,u\rangle$.
Keywords:
variational inequality, bilateral constraints, $G$-convergence of operators, convergence of solutions.
UDC:
517.9
Presented: V. I. BerdyshevReceived: 22.06.2023
Revised: 21.01.2024
Accepted: 29.01.2024
DOI:
10.31857/S2686954324010124
Cited by
