We consider variational inequalities with invertible operators \({{\mathcal {A}}}_s:W^{1,p}_0(\varOmega )\rightarrow W^{-1,p^\prime }(\varOmega )\) , \(s\in {\mathbb {N}}\) , in divergence form and constraint set \(V\subset W^{1,p}_0(\varOmega )\) defined by a measurable lower constraint \(\varphi :\varOmega \rightarrow \overline{{\mathbb {R}}}\) and a measurable upper constraint \(\psi :\varOmega \rightarrow \overline{\mathbb R}\) , where \(\varOmega \) is a nonempty bounded open set in \({\mathbb {R}}^n\) ( \(n\geqslant 2\) ) and \(p>1\) . We assume that the sequence \(\{{\mathcal A}_s\}\) G-converges to an invertible operator \({\mathcal {A}}:W^{1,p}_0(\varOmega )\rightarrow W^{-1,p^\prime }(\varOmega )\) . In addition, we assume that the set \(\{\varphi =\psi \}\) has nonempty interior, the measure of the intersection of the boundary of the set \(\{\varphi =\psi \}\) with \(\varOmega \) is zero, and there exist functions \(\bar{\varphi },\bar{\psi }\in W^{1,p}_0(\varOmega )\) such that \(\varphi \leqslant {\bar{\varphi }}\leqslant {\bar{\psi }}\leqslant \psi \) a.e. in \(\varOmega \) and the measure of the set \(\{\varphi \ne \psi \}\setminus \{{\bar{\varphi }}\ne {\bar{\psi }}\}\) is zero. Under these assumptions, we prove that the solutions of the considered variational inequalities converge weakly in \(W^{1,p}_0(\varOmega )\) to the solution of a similar variational inequality with the operator \({\mathcal {A}}\) and the constraint set V. We show that there is a fundamental difference between the considered case and the previously studied case where the measure of the set \(\{\varphi =\psi \}\) is zero.
