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

We consider variational inequalities with invertible operators ${\mathcal{A}}_s: W_0^{\mathrm{1,}p}\left(\Omega \right)\to W^{-\mathrm{1,}{p}^{\text{'}}}\left(\Omega \right), s\in \mathbb{N}$ in divergence form and constraint set $V=\left\{v\in W_0^{\mathrm{1,}p}\left(\Omega \right): \phi \le v\le \psi a.e. in \Omega \right\}$ where Ω is a nonempty bounded open set in ${ℝ}^n\left(n\ge 2\right)$, p > 1 and $\phi ,\psi : \Omega \to \overline{ℝ}$ are measurable functions. Under the assumptions that the operators ${\mathcal{A}}_s$ G-converge to an invertible operator $\mathcal{A}: W_0^{\mathrm{1,}p}\left(\Omega \right)\to W^{-\mathrm{1,}{p}^{\text{'}}}\left(\Omega \right)$, int {φ = ψ} ≠ ∅, $meas\left(\partial \left\{\phi =\psi \right\}\cap \Omega \right)=0$ and there exist functions $\overline{\phi }, \overline{\psi }\in W_0^{\mathrm{1,}p}\left(\Omega \right)$ such that $\phi \le \overline{\phi }\le \overline{\psi }\le \psi$ a.e. in Ω and $meas\left(\left\{\phi \ne \psi \right\}\backslash \left\{\overline{\phi }\ne \overline{\psi }\right\}\right)=0$ we establish the weak convergence in $W_0^{\mathrm{1,}p}\left(\Omega \right)$ of the solutions u_s of the specified variational inequalities to the solution u of a similar variational inequality with the operator $\mathcal{A}$ and the constraint set V. The fundamental difference between the considered case and the previously studied case, where meas {φ = ψ} = 0 is that, in general, the functionals ${\mathcal{A}}_s{u}_s$ do not converge to $\mathcal{A}u$ even weakly in W^{–1, p'} (Ω) and the energy integrals $<{\mathcal{A}}_s{u}_s, u_s>$ do not converge to $<\mathcal{A}u, u>$.