Elliptic Variational Inequalities with Discontinuous Multi-Valued Lower Order Terms

Abstract

AbstractWe consider multi-valued elliptic variational inequalities for operators of the formu ↦ Au + ∂2ψ(u, u),where A is a second order elliptic operator of Leray-Lions type, and u ↦ ∂2ψ(u, u) is a multi-valued lower order term that may neither be lower nor upper semicontinuous, see e.g., Figure 2. More precisely, the lower order term is generated by some function (r, s) → ψ(r, s) with ψ : ℝ × ℝ → ℝ that is locally Lipschitz with respect to its second variable s for each r ∈ ℝ, and ∂2ψ(r, s) denotes Clarke’s generalized gradient of s ↦ ψ(r, s). The novelty of this paper is that the multifunction r ↦ ∂2ψ(r, s) may discontinuously depend on r in a certain specified way, which gives rise to the new class of discontinuous multi-valued lower order terms s ↦ ∂2ψ(s, s). Though rich in structure, the characteristic features of this new class of multi-valued lower order terms can easily be described and verified. Our main goal is to provide an analytical framework and to prove existence and comparison results for this new class of multi-valued variational inequalities that includes the theory of variational-hemivariational inequalities as special case.