Parabolic variational inequalities with generalized reflecting directions

Abstract

Abstract:We study, in a Hilbert framework, some abstract parabolic variational inequalities, governed by reflecting subgradients with multiplicative perturbation, of the following type:where A is a linear self-adjoint operator, @' is the subdifferential operator of a proper lower semicontinuous convex function ' defined on a suitable Hilbert space, and ‚ is the perturbing term which acts on the set of reflecting directions, destroying the maximal monotony of the multivalued term. We provide the existence of a solution for the above Cauchy problem. Our evolution equation is accompanied by examples which aim to (systems of) PDEs with perturbed reflection.