Tykhonov Well-posedness of a Heat Transfer Problem with Unilateral Constraints

TL;DR

This paper investigates the well-posedness of a heat transfer problem with unilateral constraints using hemivariational inequalities, Tykhonov triples, and perturbation analysis to establish existence, stability, and convergence results.

Contribution

It introduces a novel application of Tykhonov triples to analyze the well-posedness of a heat transfer problem with unilateral constraints and boundary conditions.

Findings

Established well-posedness results for the problem and its control version.

Proved convergence of penalty approximations under perturbations.

Applied theoretical results to specific boundary value problem variants.

Abstract

We consider an elliptic boundary value problem with unilateral constraints and subdifferential boundary conditions. The problem describes the heat transfer in a domain $D\subset\R^d$ and its weak formulation is in the form of a hemivariational inequality for the temperature field, denoted by $\cP$. We associate to Problem $\cP$ an optimal control problem, denoted by $\cQ$. Then, using appropriate Tykhonov triples, governed by a nonlinear operator $G$ and a convex $\wK$, we provide results concerning the well-posedness of problems $\cP$ and $\cQ$. Our main results are Theorems 14 and 18, together with their corollaries. Their proofs are based on arguments of compactness, lower semicontinuity and pseudomonotonicity. Moreover, we consider three relevant perturbations of the heat transfer boundary valued problem which lead to penalty versions of Problem $\cP$, constructed with particular choices of $G$ and $\wK$. We prove that Theorems 14 and 18 as well as their corollaries can be applied in the study of these problems, in order to obtain various convergence results.