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First, we prove existence and comparison results for multi-valued elliptic variational inequalities involving Clarke’s generalized gradient of some locally Lipschitz functions as multi-valued term. Only by applying the definition of Clarke’s gradient it is well known that any solution of such a multi-valued elliptic variational inequality is also a solution of a corresponding variational-hemivariational inequality. The reverse is known to be true if the locally Lipschitz functions are regular in the sense of Clarke. Without imposing this kind of regularity the equivalence of the two problems under consideration is not clear at all. The main goal of this paper is to show that the equivalence still holds true without any additional regularity, which will fill a gap in the literature. Existence and comparison results for both multi-valued variational inequalities and variational-hemivariational inequalities are the main tools in the proof of the equivalence of these problems.

Equivalence of Some Multi-Valued Elliptic Variational Inequalities and Variational-Hemivariational Inequalities