The Caffarelli Alternative in Measure for the Nondivergence Form Elliptic Obstacle Problem with Principal Coefficients in VMO

TL;DR

This paper advances the understanding of obstacle problems with elliptic operators in nondivergence form having VMO coefficients, establishing fundamental theory and initial free boundary regularity results.

Contribution

It develops existence, uniqueness, regularity, and nondegeneracy results for solutions, and introduces measure stability and a measure-theoretic Caffarelli alternative for free boundary analysis.

Findings

Existence and uniqueness of solutions established.

Optimal regularity and nondegeneracy proven.

Existence of blowup limits and measure stability shown.

Abstract

We study the obstacle problem with an elliptic operator in nondivergence form with principal coefficients in VMO. 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, and we show existence of blowup limits, a basic measure stability result, and a measure-theoretic version of the Caffarelli alternative proven in Caffarelli's 1977 paper "The regularity of free boundaries in higher dimensions."