Undergrad Thesis–Regularity of Free Boundaries

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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 C^{1,\alpha}.”  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 \Omega, such that u is harmonic on \Omega^+ = \{x: u(x) > 0\} and  \Omega^- = \text{interior}({x: u(x) \le 0}).  How smooth is the free boundary F = \partial \Omega^+ \cap \Omega?  We assume that in a neighborhood of a point, F is a Lipschitz graph, and we show that in fact it is the graph of a C^{1,\alpha} function f.

The main idea is this:  We prove that in a ball around a boundary point x_0, u is increasing in a cone of directions of some angle.  Thus, there is a cone based at x_0 which is contained in \Omega^+.  Using estimates which compare two solutions, we show that if u is increasing in a cone of directions near x_0, then on a smaller neighborhood of x_0, u is increasing in a wider cone of directions.  Applying this observation iteratively along with the corresponding estimates shows that f is differentiable at any point x_0, and the derivative is in fact Hölder continuous.

Art of a Mathematician