Harmonic Measure - Geometric and Analytic Points of View
This book is based on a series of lectures given by Carlos Kenig in March 2000 at the University of Arkansas. The Dirichlet problem for the Laplace equation can be solved (without any regularity restriction on the domain) by integration with respect to the harmonic measure of the domain. On good domains, the harmonic measure may have density with respect to the surface measure; this is the Poisson kernel. It requires hard analysis to relate regularity properties of the harmonic measure to the geometry of the domain.
The main goal of this book is to present recent deep results by Kenig and Toro. They completely settle two questions. The first one is when the harmonic measure is asymptotically locally doubling. This roughly means that the ratio μ(B(x,σ r))/μ(B(x,r)) tends to σn as r approaches zero. The answer is in terms of the flatness of the domain (vanishing Reichenberg flatness). The second question is when the Poisson kernel exists and has VMO logarithm (VMO means the well known space of functions with vanishing mean oscillation). The answer is in terms of the flatness and asymptotic behaviour of the surface (perimeter) measure and the condition is called the vanishing chord arc. All main results are in general dimension. The presentation is mostly developed in two stages, first the scheme of the proof is outlined and then the exposition proceeds to complete details. This helps in understanding the idea of the proof. The results mentioned above mainly occur as natural end points of previously known results of Kellogg, Alt, Caffarelli, Dahlberg, Jerison, Kenig and others. These results, as well as two-dimensional results, are presented in the form of a survey. Some background is included to make the exposition more self-contained and accessible.