Convergence Results for Elliptic Variational-Hemivariational Inequalities

Abstract

We consider an elliptic variational-hemivariational inequality 𝓟 in a reflexive Banach space, governed by a set of constraints K, a nonlinear operator A, and an element f. We associate to this inequality a sequence {𝓟n} of variational-hemivariational inequalities such that, for each n ∈ ℕ, inequality 𝓟n is obtained by perturbing the data K and A and, moreover, it contains an additional term governed by a small parameter εn. The unique solvability of 𝓟 and, for each n ∈ ℕ, the solvability of its perturbed version 𝓟n, are guaranteed by an existence and uniqueness result obtained in literature. Denote by u the solution of Problem 𝓟 and, for each n ∈ ℕ, let un be a solution of Problem 𝓟n. The main result of this paper states the strong convergence of un → u in X, as n → ∞. We show that the main result extends a number of results previously obtained in the study of Problem 𝓟. Finally, we illustrate the use of our abstract results in the study of a mathematical model which describes the contact of an elastic body with a rigid-deformable foundation and provide the corresponding mechanical interpretations.

Keywords

• variational-hemivariational inequality
• penalty operator
• mosco convergence
• internal approximation
• tykhonov well-posedness
• contact problem
• 47j20
• 49j40
• 49j45
• 35m86
• 74m10
• 74m15