Signorini type variational inequality with state-dependent discontinuous multi-valued boundary operators

Abstract

We consider an elliptic variational inequality of Signorini type in a bounded domain Ω⊂RN of the form: find u∈K and ξ∈∂2β(γu,γu) such that 〈Au,v−u〉+∫ΓSξ(γv−γu)dσ≥0,∀v∈K, where A is a quasilinear elliptic operator in divergence form, K is the nonempty, closed, convex set representing the thin obstacle (Signorini problem) on ΓS⊂∂Ω, and γ:W1,p(Ω)→Lp(∂Ω) denotes the trace operator. The multi-valued boundary operator is generated by the multifunction s↦∂2β(s,s), where β:R×R→R is a function such that s↦β(r,s) is supposed to be locally Lipschitz for all r∈R, while r↦β(r,s) is allowed to be discontinuous, and ∂2β(r,s) stands for Clarke’s generalized gradient of β with respect to its second argument. The novelty of this paper is that the multifunction r↦∂2β(r,s) may discontinuously depend on r in a certain specified way, which gives rise to a new class of discontinuous, nonmonotone, multi-valued boundary operators that is rich in structure to cover a wide range of multi-valued constitutive laws on the contact surface ΓS such as, e.g., multi-valued adhesion and friction laws that are not necessarily of subdifferential type. Our main goal is to provide an analytical framework to prove existence, enclosure and comparison results for this new class of multi-valued boundary variational inequalities that includes the theory of variational–hemivariational inequalities as special case.