Nonlinear Variational Inequalities with Bilateral Constraints Coinciding on a Set of Positive Measure

Abstract

We consider variational inequalities with invertible operators \({{\mathcal{A}}_{s}}{\text{:}}~\,W_{0}^{{1,p}}\left( {{\Omega }} \right) \to {{W}^{{ - 1,p'}}}\left( {{\Omega }} \right),\) \(s \in \mathbb{N},\) in divergence form and with constraint set \(V = \{ {v} \in W_{0}^{{1,p}}\left( {{\Omega }} \right){\text{: }}\varphi \leqslant {v} \leqslant \psi ~\) a.e. in \({{\Omega }}\} ,\) where \({{\Omega }}\) is a nonempty bounded open set in \({{\mathbb{R}}^{n}}\) \(\left( {n \geqslant 2} \right)\), p > 1, and \(\varphi ,\psi {{:\;\Omega }} \to \bar {\mathbb{R}}\) are measurable functions. Under the assumptions that the operators \({{\mathcal{A}}_{s}}\) G-converge to an invertible operator \(\mathcal{A}{\text{: }}W_{0}^{{1,p}}\left( {{\Omega }} \right) \to {{W}^{{ - 1,p'}}}\left( {{\Omega }} \right)\), \({\text{int}}\left\{ {\varphi = \psi } \right\} \ne \varnothing ,\) \({\text{meas}}\left( {\partial \left\{ {\varphi = \psi } \right\} \cap {{\Omega }}} \right)\) = 0, and there exist functions \(\bar {\varphi },\bar {\psi } \in W_{0}^{{1,p}}\left( {{\Omega }} \right)\) such that \(\varphi \leqslant \overline {\varphi ~} \leqslant \bar {\psi } \leqslant \psi \) a.e. in \({{\Omega }}\) and \({\text{meas}}\left( {\left\{ {\varphi \ne \psi } \right\}{{\backslash }}\left\{ {\bar {\varphi } \ne \bar {\psi }} \right\}} \right) = 0,\) we establish that the solutions u_s of the variational inequalities converge weakly in \(W_{0}^{{1,p}}\left( {{\Omega }} \right)\) 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 \({\text{meas}}\left\{ {\varphi = \psi } \right\} = 0\) is that, in general, the functionals \({{\mathcal{A}}_{s}}{{u}_{s}}\) do not converge to \(\mathcal{A}u\) even weakly in \({{W}^{{ - 1,p'}}}\left( {{\Omega }} \right)\) and the energy integrals \(\langle {{\mathcal{A}}_{s}}{{u}_{s}},{{u}_{s}}\rangle \) do not converge to \(\langle \mathcal{A}u,u\rangle \).