Two phase free boundary problem for harmonic measure: blow-up techniques

The first lecture will briefly cover prerequisites from Mattila's "Geometry of Sets and Measures" including: rectifiability and tangent measures. It will also give an overview of the types of results one could hope for relating the behavior of measures corresponding to a Dirichlet problem and the geometry of the boundary.

We will proceed to discuss the two phase harmonic measure free boundary problem, with an initial discussion of the special case of the complex plane. We will then study the blow-up analysis and techniques from Kenig-Toro 2003, Kenig-Toro 2006, and Kenig-Preiss-Toro 2009.

Date and time info
Tuesdays, 09.30-11.00

Keywords
harmonic measure, geometric measure theory, blow-up analysis, free boundary problems

Prerequisites
Real analysis, a first course in PDE