Reifenberg Flatness of Free Boundaries in Obstacle Problems with VMO Ingredients

TL;DR

This paper investigates the regularity of free boundaries in obstacle problems with divergence form elliptic operators having VMO coefficients, establishing foundational existence, uniqueness, and regularity results.

Contribution

It develops the basic theory for obstacle problems with VMO coefficients, enabling further study of free boundary regularity in this context.

Findings

Established existence and uniqueness of solutions

Proved optimal regularity and nondegeneracy of solutions

Laid groundwork for free boundary regularity analysis

Abstract

We study the obstacle problem with an elliptic operator in divergence form. We develop all of the basic theory of existence, uniqueness, optimal regularity, and nondegeneracy of the solutions. These results, in turn, allow us to begin the study of the regularity of the free boundary in the case where the coefficients are in VMO.