Abstract: I will talk about two related topics of elliptic PDEs: quantitative unique continuation and harmonic measure. Unique continuation property is a fundamental property of harmonic functions: If a harmonic function vanishes to infinite order at a point, it must vanish everywhere. In the first part of the talk, I will explain how to use the local growth rate of a harmonic function to estimate the size of its singular set .
Let be a bounded domain and be fixed. A Brownian traveller starting from must exit the domain, and the harmonic measure is a probability measure on the boundary which characterizes the distribution where exits the domain from. It also gives a simple representation formula for solutions to the boundary value problem of the Laplacian in . Thus the harmonic measure, and more generally the elliptic measure, carries a lot of information about the boundary behavior of these solutions. In the second part of the talk, I will explain the properties of harmonic/elliptic measures in non-smooth domains, and in particular how they are related to the natural surface measure of the boundary.