A Penalty Method for Elliptic Variational–Hemivariational Inequalities

We consider an elliptic variational–hemivariational inequality P in a real reflexive Banach space, governed by a set of constraints K. Under appropriate assumptions of the data, this inequality has a unique solution u∈K. We associate inequality P to a sequence of elliptic variational–hemivariational inequalities {Pn}, governed by a set of constraints K˜⊃K, a sequence of parameters {λn}⊂R+, and a function ψ. We prove that if, for each n∈N, the element un∈K˜ represents a solution to Problem Pn, then the sequence {un} converges to u as λn→0. Based on this general result, we recover convergence results for various associated penalty methods previously obtained in the literature. These convergence results are obtained by considering particular choices of the set K˜ and the function ψ. The corresponding penalty methods can be applied in the study of various inequality problems. To provide an example, we consider a purely hemivariational inequality that describes the equilibrium of an elastic membrane in contact with an obstacle, the so-called foundation.