Toward the L1-theory of degenerate anisotropic elliptic variational inequalities

Abstract

We consider nonlinear elliptic second-order variational inequalities with degenerate (with respect to the spatial variable) and anisotropic coefficients and L¹-data. We study the cases where the set of constraints belongs to a certain anisotropic weighted Sobolev space and to a larger function class. In the first case, some new properties of T-solutions and shift T-solutions of the investigated variational inequalities are established. Moreover, the notion of W^1,1-regular T-solution is introduced, and a theorem of existence and uniqueness of such a solution is proved. In the second case, we introduce the notion of T-solution of the variational inequalities under consideration and establish conditions of existence and uniqueness of such a solution.