Convergence analysis for double phase obstacle problems with multivalued convection term
The authors consider the following double phase problem with a multivalued convection term and obstacle effect \[ (P) \quad \begin{cases} - \mbox{div}\, (|\nabla u|^{p-2}\nabla u+\mu(x)|\nabla u|^{q-2}\nabla u) \in f(x,u,\nabla u) \quad \mbox{in }\Omega, \\ u(x)\leq \Phi(x)\quad \mbox{in }\Omega,\\ u=0 \quad \mbox{on }\partial \Omega, \, 1<p<q<N, \end{cases} \] where \(\Omega \subseteq \mathbb{R}^N\) is a bounded domain with a Lipschitz boundary \(\partial \Omega\), \(\mu :\overline{\Omega}\to[0,\infty)\) is Lipschits continuous, \(f :\Omega \times \mathbb{R}\times \mathbb{R}^N\to 2^{\mathbb{R}}\) and \(\Phi:\Omega \to [0,\infty)\) satisfying suitable assumptions. The authors consider a family of approximating problems corresponding to \((P)\), and obtain a convergence theorem which establishes that the solution set of \((P)\) can be approximated by the solutions of such a family of auxiliary ``perturbation'' problems. The methods used are based on monotone operator theory and their adaptation to the specific problems for a successful application.
scientific article; zbMATH DE number 7363131
scientific article