I have been working under Professor Tatiana Toro to revise and exposit Luis Caffarelli’s paper, “A Harnack Inequality Approach to the Regularity of Free Boundaries: Part I: Lipschitz Free Boundaries are .” This paper was extremely important in the study of solutions to free boundary problems in elliptic partial differential equations, but needed to be made more accessible.
The basic setup is this. We are given a function continuous u on a region , such that is harmonic on and . How smooth is the free boundary ? We assume that in a neighborhood of a point, is a Lipschitz graph, and we show that in fact it is the graph of a function .
The main idea is this: We prove that in a ball around a boundary point , is increasing in a cone of directions of some angle. Thus, there is a cone based at which is contained in . Using estimates which compare two solutions, we show that if is increasing in a cone of directions near , then on a smaller neighborhood of , is increasing in a wider cone of directions. Applying this observation iteratively along with the corresponding estimates shows that is differentiable at any point , and the derivative is in fact Hölder continuous.