On the Tykhonov Well-Posedness of an Antiplane Shear Problem

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 ${{{\mathcal {P}}}}$. We associated with problem ${{{\mathcal {P}}}}$ a boundary optimal control problem, denoted by ${{{\mathcal {Q}}}}$. For Problem ${{{\mathcal {P}}}},$ we introduce the concept of well-posedness and for Problem ${{{\mathcal {Q}}}}$ we introduce the concept of weakly and weakly generalized well-posedness, both associated with appropriate Tykhonov triples. Our main results are Theorems 5 and 16. Theorem 5 provides the well-posedness of Problem ${{{\mathcal {P}}}}$ and, as a consequence, the continuous dependence of the solution with respect to the data. Theorem 16 provides the weakly generalized well-posedness of Problem ${{{\mathcal {Q}}}}$ 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.