Well-posedness and convergence results for elliptic hemivariational inequalities

Abstract

We consider an elliptic hemivariational inequality in a real reflexive Banach space X which, under appropriate assumptions on the data, has a unique solution u ∈ X. We recall the concepts of wellposedness in the sense of Tykhonov and Levitin-Polyak for this inequality, and then we extend these concepts by introducing new well-posedness concepts, constructed with a larger set of approximating sequences. We also prove that, under additional assumptions, these new well-posedness concepts are optimal in the sense that all the sequences of elements of X which converge to the solution u are approximating sequences. This result, presented in Theorem 4.1, provides necessary and sufficient conditions for any sequence {un} ⊂ X which guarantees that it converges to u and, therefore, it represents a convergence criterion to the solution of the hemivariational inequality. This criterion can be used in various applications. To provide an example, we illustrate its use in the study of a penalty method associated to an elliptic hemivariational inequality which describes the equilibrium of an elastic membrane in contact with a obstacle, the so-called foundation.