Regularity Theory for Thin Obstacle Problems

Abstract

The goal of this PhD thesis is to collect the results of the author in the study of thin obstacle problems. We start by giving an introduction to the Signorini or thin obsta\-cle problem, summarizing some of the most relevant currently known results. The next chapters correspond each to one paper by the author (and collaborators). Thus, we start by studying the regularity of solutions for the fully nonlinear thin obstacle problem, to then move to the study of the free boundary for general fractional obstacle problems with drift, in the critical regime. This is followed by a regularity result for minimizers of the perimeter with lower dimensional obstacles. Finally, the last two chapters focus on the standard thin obstacle problem (and its fractional counter-part) and fine regularity and generic regularity properties for the free boundary.