Regularity theory for the obstacle problem

Abstract

In this work, we introduce and study a classical free boundary problem known as the obstacle problem. This problem serves as a foundational example in the broader theory of free boundary problems, where part of the solution involves determining an unknown interface or region. We begin by presenting the classical formulation of the obstacle problem, which arises naturally in various physical and geometric contexts, such as elasticity, fluid dynamics, and potential theory. We then explore the key theoretical aspects of the problem, focusing on the existence, uniqueness, and regularity of solutions. Special attention is given to the structure and behavior of the free boundary, the interface separating the active and inactive regions which plays a central role in understanding the qualitative features of the solution. To analyze the behavior of the solution near the free boundary, we employ the method of blow-ups, a powerful technique that allows for the study of local properties by rescaling the problem around singular points. This approach provides deep insights into the regularity and classification of free boundary points, distinguishing between regular and singular behavior and leading to a better understanding of the geometry of the free boundary. Overall, this work offers a rigorous introduction to the obstacle problem, combining classical theory with modern analytical tools to examine one of the most important and illustrative problems in the study of partial differential equations and variational inequalities.