Constrained Variational-Hemivariational Inequalities on Nonconvex Star-Shaped Sets

Stanisław Migórski,Long Fengzhen

doi:10.3390/math8101824

Abstract

In this paper, we study a class of constrained variational-hemivariational inequality problems with nonconvex sets which are star-shaped with respect to a certain ball in a reflexive Banach space. The inequality is a fully nonconvex counterpart of the variational-hemivariational inequality of elliptic type since it contains both, a convex potential and a locally Lipschitz one. Two new results on the existence of a solution are proved by a penalty method applied to a variational-hemivariational inequality penalized by the generalized directional derivative of the distance function of the constraint set. In the first existence theorem, the strong monotonicity of the governing operator and a relaxed monotonicity condition of the Clarke subgradient are assumed. In the second existence result, these two hypotheses are relaxed and a suitable hypothesis on the upper semicontinuity of the operator is adopted. In both results, the penalized problems are solved by using the Knaster, Kuratowski, and Mazurkiewicz (KKM) lemma. For a suffciently small penalty parameter, the solution to the penalized problem solves also the original one. Finally, we work out an example on the interior and boundary semipermeability problem that ilustrate the applicability of our results.

Highlights

We are initially motivated by the investigation of the class of variational-hemivariational inequalities considered in [1]
Particular forms of problem (1) contain various formulations investigated in the literature: the elliptic variational inequalities of the first and second kind, see [2,3,4,5], the elliptic hemivariational inequalities, see [6,7,8,9], and the elliptic equations, see [6,10,11]
Mathematics 2020, 8, 1824 corresponding to problem (1) and its variants can be treated by a fixed point technique, see [1,5]

Summary

Introduction

We are initially motivated by the investigation of the class of variational-hemivariational inequalities considered in [1]. We treat the counterpart of problem (1) where the set K represents a set of admissible constraints which is star-shaped with respect to a certain ball in V Note that for this class of nonconvex sets, some particular versions have been studied earlier in Section 7.4 of [8] if j = 0, in Section 7.3 of [8,12] if φ = 0, and Section 7.2 of [8] when φ = j = 0. The second novelty is to study the constrained variational-hemivariational inequality on star-shaped sets without hypothesis on the strong monotonicity of the operator and without the relaxed monotonicity condition of the generalized subgradient.

Basic Material

Formulation of the Problem

Proof of Theorem 1

Second Existence Result

Semipermeability Model

Conclusions