Extremal solutions of multi-valued variational inequalities in plane exterior domains

Abstract

Let Ω=R2∖B(0,1)‾ be the exterior of the closed unit disc in the plane. In this paper we prove existence and enclosure results of multi-valued variational inequalities in Ω of the form: Find u∈K and η∈F(u) such that〈−Δu,v−u〉≥〈aη,v−u〉,∀v∈K, where K is a closed convex subset of the Hilbert space X=D1,20(Ω) which is the completion of Cc∞(Ω) with respect to the ‖∇⋅‖2,Ω-norm. The lower order multi-valued operator F is generated by an upper semicontinuous multi-valued function f:R→2R∖{∅}, and the (single-valued) coefficient a:Ω→R+ is supposed to decay like |x|−2−α with α>0. Unlike in the situation of higher-dimensional exterior domain, that is RN∖B(0,1)‾ with N≥3, the borderline case N=2 considered here requires new tools for its treatment and results in a qualitatively different behaviour of its solutions. We establish a sub-supersolution principle for the above multi-valued variational inequality and prove the existence of extremal solutions. Moreover, we are going to show that classes of generalized variational-hemivariational inequalities turn out to be merely special cases of the above multi-valued variational inequality.