Existence, comparison, and convergence results for a class of elliptic hemivariational inequalities

TL;DR

This paper investigates elliptic boundary hemivariational inequalities related to heat conduction, establishing existence, comparison, asymptotic behavior, and continuous dependence of solutions, with applications to nonmonotone boundary conditions.

Contribution

It introduces new existence results using pseudomonotone operator theory and analyzes solution behavior under various boundary conditions and parameters.

Findings

Existence of solutions proven using pseudomonotone operator theory.

Comparison results for solutions under different conditions.

Asymptotic behavior of solutions as heat transfer coefficient increases.

Abstract

In this paper we study a class of elliptic boundary hemivariational inequalities which originates in the steady-state heat conduction problem with nonmonotone multivalued subdifferential boundary condition on a portion of the boundary described by the Clarke generalized gradient of a locally Lipschitz function. First, we prove a new existence result for the inequality employing the theory of pseudomonotone operators. Next, we give a result on comparison of solutions, and provide sufficient conditions that guarantee the asymptotic behavior of solution, when the heat transfer coefficient tends to infinity. Further, we show a result on the continuous dependence of solution on the internal energy and heat flux. Finally, some examples of convex and nonconvex potentials illustrate our hypotheses.