A Class of Nonlinear Elliptic Variational Inequalities: Qualitative Properties and Existence of Solutions

Abstract

We study a class of nonlinear elliptic variational inequalities in divergence form. In the recent paper [6], we obtained results on the local control of essential infimum and supremum of solutions of quasilinear elliptic equations, and here we extend this point of view to the case of variational inequalities. It implies a new qualitative property of solutions in W1,p(Ω) which we call ``jumping over the control obstacle.'' Using the Schwarz symmetrization technique, we give an existence and symmetrization theorems in W1,p0(Ω) ∩ L∞(Ω) which agree completely with previous qualitative results. Also we consider generating singularities of weak solutions in W1,p(Ω) of variational inequalities.