Obstacle type problems in Orlicz-Sobolev spaces

Tese de doutoramento, Matemática (Física Matemática e Mecânica dos Meios Contínuos), Universidade de Lisboa, Faculdade de Ciências, 2013This thesis consists of four chapters. In the first chapter we study the regularity of solutions for a class of elliptic problems in Orlicz-Sobolev spaces. In particular, we see that bounded weak solutions of Au := div a(x; jruj)ru _ = f(x); x 2 ; where Ω ⊂ Rn is a bounded domain, for an appropriate a and f are C1 α regular. Using Lewy-Stampacchia inequalities for one obstacle problem we derive C1 α regularity results (both locally and up to the boundary) for the solution of a quasilinear obstacle problem. In the second chapter we prove Lewy-Stampacchia inequalities in abstract form for two obstacles problem and for N-membranes problem. Applying those inequalities we derive C1 α regularity results (both locally and up to the boundary) for A(x)-obstacle problem with two obstacles and for N-membranes problem. As another application of Lewy-Stampacchia inequalities, we study a quasivariational problem related to a stochastic switching game. We prove, that the problem admits at least a maximal and a minimal solution. In the third chapter we extend the regularity of the free boundary of the obstacle problem to a class of heterogeneous quasilinear degenerate elliptic operators (including p(x)-Laplacian). We prove that the free boundary is a porous set and hence has Lebesgue measure zero. We also show that the (n - 1)-dimensional Hausdorff measure of the free boundary is finite (for p(x) > 2), which yields, in particular, that up to a negligible singular set, the free boundary is the union of at most a countable family of C1 hypersurfaces. Finally, in the chapter four of the thesis, after homogenizing the Dirichlet problem for A(x)-Laplacian in Orlicz-Sobolev spaces, we study the homogenization of the A(x)-obstacle problem, then prove convergence of the coincidence sets.Fundação para a Ciência e a Tecnologia (FCT, SFRH/BD/40819/2007