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

TL;DR

This paper studies a complex boundary value problem involving a double phase operator, nonlinear convection, multivalued terms, and an obstacle constraint, proving existence and compactness of solutions under broad conditions.

Contribution

It introduces a novel analysis of implicit obstacle problems with a double phase operator and multivalued nonlinearities, establishing solution existence and weak compactness.

Findings

Solution set is nonempty.

Solutions are weakly compact.

Applicable fixed point and variational methods are demonstrated.

Abstract

In this paper we consider a mixed boundary value problem with a nonhomogeneous, nonlinear differential operator (called double phase operator), a nonlinear convection term (a reaction term depending on the gradient), three multivalued terms and an implicit obstacle constraint. Under very general assumptions on the data, we prove that the solution set of such implicit obstacle problem is nonempty (so there is at least one solution) and weakly compact. The proof of our main result uses the Kakutani-Ky Fan fixed point theorem for multivalued operators along with the theory of nonsmooth analysis and variational methods for pseudomonotone operators.