Nonlinear variational inequalities with bilateral constraints coinciding on a set of positive measure

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 $\langle\mathcal Au,u\rangle$.