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

TL;DR

This paper proves the existence of solutions for a complex double phase obstacle problem with multivalued convection and mixed boundary conditions, and analyzes the convergence of approximating solutions to the original problem.

Contribution

It introduces a novel existence theorem for a double phase obstacle problem with multivalued convection and establishes convergence of approximations to the solution set.

Findings

Existence of solutions under general assumptions.

Convergence of approximating solutions to the actual solution set.

Use of surjectivity theorem and Moreau-Yosida approximation methods.

Abstract

In this paper, we consider a mixed boundary value problem with a double phase partial differential operator, an obstacle effect and a multivalued reaction convection term. Under very general assumptions, an existence theorem for the mixed boundary value problem under consideration is proved by using a surjectivity theorem for multivalued pseudomonotone operators together with the approximation method of Moreau-Yosida. Then, we introduce a family of the approximating problems without constraints corresponding to the mixed boundary value problem. Denoting by $\mathcal S$ the solution set of the mixed boundary value problem and by $\mathcal S_n$ the solution sets of the approximating problems, we establish the following convergence relation \begin{align*} \emptyset\neq w\text{-}\limsup\limits_{n\to\infty}{\mathcal S}_n=s\text{-}\limsup\limits_{n\to\infty}{\mathcal S}_n\subset \mathcal S, \end{align*} where $w$-$\limsup_{n\to\infty}\mathcal S_n$ and $s$-$\limsup_{n\to\infty}\mathcal S_n$ stand for the weak and the strong Kuratowski upper limit of $\mathcal S_n$, respectively.