Introductory talk: Introduction to Layer Potentials. Main talk: A Singular Integral Approach to Two-Phase Free Boundary Problems

Introductory talk: We introduce the single and double layer potentials through Greens second identity. Roughly speaking, these are the best guesses to the solution of the Neumann and Dirichlet boundary value problems. We will also discuss the "jump relations" for the layer potentials and the important solvability results connected to the layer potential method. Finally, we give an indication of how the single layer can be used to investigate two-phase free boundary problems.

 

Main talk: We will put the ideas of the first talk to use in the study of two-phase free boundary problems. In particular, we will sketch how quantitative control on the mean oscillation of Poisson kernel for a domain and its exterior yields similar control on the mean oscillation of the unit normal vector. This is an extension of the work of Kenig and Toro. If time permits, we will discuss how to draw the conclusion of Reifenberg flatness from small mean oscillation of the unit normal vector. This is joint work with Max Engelstein and Steve Hofmann.