Uma introdução aos problemas do tipo obstáculo

Abstract

In this study, based on the book “Regularity of Free Boundaries in Obstacle-Type Problems” by Arshak Petrosyan, Henrik Shahgholian, and Nina Uraltseva, [22], we investigate the geometric and analytical properties of solutions to obstacle-type problems. In the problems to be addressed below, the equations that model diffusion phenomena exhibit discontinuity jumps along a set that depends on the solution itself, which we call the free boundary. Initially, we focus on simpler models where solutions do not change sign. Subsequently, we expand our analysis to more complex scenarios, considering cases with sign changes, where we study, among other things, estimates of local and global regularity for solutions. We explore various geometric features of the free boundary, including the non-degeneracy of solutions and the Hausdorff measure of this set. As a result, we establish Lipschitz and C1 regularity for the free boundary.