Free boundary regularity for harmonic measures and Poisson kernels

Carlos E. Kenig and Tatiana Toro

Review from Zentralblatt MATH:

The authors study the relationship between the geometry of ``hypersurface like'' subsets of Euclidean space and the properties of the measure they support. It is shown that if the doubling constant of a measure on ${\Bbb R}^{n+1}$ is close to the doubling constant of $n$-dimensional Lebesgue measure, then its support is well approximated by $n$-dimensional affine spaces, if the support is relatively flat to begin with. They also investigate how the ``weak'' regularity of the Poissson kernel of a domain determines the geometry of its boundary. Under some additional hypothesis they prove that the oscillation of the Poisson kernel controls the oscillation of the unit normal vector.

Reviewed by Olof Svensson

Keywords: geometric measure theory; regularity of the Poisson kernel; oscillation of the Poisson kernel

Classification (MSC2000): 31B25 28A75 35R35

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