On the Tykhonov Well-posedness of an Antiplane Shear Problem

TL;DR

This paper investigates the stability and well-posedness of a static antiplane shear problem with friction, establishing conditions under which solutions depend continuously on data and introducing new concepts of well-posedness for related control problems.

Contribution

It introduces the concept of Tykhonov well-posedness for an antiplane shear problem and its control formulation, proving stability results and providing mechanical interpretations.

Findings

Proves the well-posedness of the shear problem with continuous data dependence.

Establishes weakly and weakly generalized well-posedness for the control problem.

Provides mechanical interpretation of the mathematical results.

Abstract

We consider a boundary value problem which describes the frictional antiplane shear of an elastic body. The process is static and friction is modeled with a slip-dependent version of Coulomb's law of dry friction. The weak formulation of the problem is in the form of a quasivariational inequality for the displacement field, denoted by $\cP$. We associated to problem $\cP$ a boundary optimal control problem, denoted by $\cQ$. For Problem $\cP$ we introduce the concept of well-posedness and for Problem $\cQ$ we introduce the concept of weakly and weakly generalized well-posedness, both associated to appropriate Tykhonov triples. Our main result are Theorems \ref{t1} and \ref{t2}. Theorem \ref{t1} provides the well-posedness of Problem $\cP$ and, as a consequence, the continuous dependence of the solution with respect to the data. Theorem \ref{t2} provides the weakly generalized well-posedness of Problem $\cQ$ and, under additional hypothesis, its weakly well posedness. The proofs of these theorems are based on arguments of compactness, lower semicontinuity, monotonicity and various estimates. Moreover, we provide the mechanical interpretation of our well-posedness results.