Some special problems in elliptic and parabolic variational inequalities

Abstract: This Licentiate Thesis is devoted above all to the investigation of variational inequalities. Chapter 1 deals with linear elliptic variational inequalities, where the operator is degenerated or singular, which involves the use of some weighted Sobolev spaces. It is shown in several examples how to interprete the (weak) solution of such variational inequality, if it is regular. In the next chapters, parabolic variational inequalities or equations on non-cylindrical domains are considered and the existence of a (weak) solution is proved by a generalization of the so-called method of Rothe. Chapter 2 is devoted to nonlinear parabolic inequalities with strongly elliptic part, while Chapter 3 deals with a linear parabolic equation, in which some singularities appear at du/dt as well as in the elliptic part, which involves the use of some weighted Sobolev spaces. In Chapter 4, the approach of Chapter 3 is extended from equations to linear singular parabolic inequalities.