Double phase obstacle problems with multivalued convection and mixed boundary value conditions (Q2093453)

Let \(\Omega \subset \mathbb{R}^N\) (\(N>2\)) be a bounded domain with Lipschitz continuous boundary \(\partial \Omega\). The authors study a mixed boundary value obstacle problem driven by a double phase differential operator of the form \[ D_\mu(u)=-\operatorname{div}\left(|\nabla u|^{p-2}\nabla u+\mu(x) |\nabla u|^{q-2}\nabla u\right) \] where \(1<p<q<N\) and \(\mu:\Omega \to [0,+\infty)\) is a suitable bounded function. The main features of the problem are the presence of: \begin{itemize} \item[1.] a multivalued convection term; \item[2.] an obstacle restriction; \item[3.] a multivalued boundary condition. \end{itemize} The authors discuss the nonemptiness, boundedness and closedness of the solution set to the mixed boundary value obstacle problem, by combining a surjectivity theorem for multivalued pseudomonotone operators and the approximation method of Moreau-Yosida. Furthermore, the authors obtain a critical convergence result establishing that the solution set of the obstacle problem can be approximated by the solution sets of auxiliary approximating problems without obstacle, in the sense of Kuratowski.