Some questions concerning harmonic measure. (English) Zbl 0792.30005

Bishop, Christopher J.

Dahlberg, B. (ed.) et al., Partial differential equations with minimal smoothness and applications. Proceedings of an IMA Participating Institutions (PI) conference, held Chicago, IL, USA, March 21-25, 1990. New York etc.: Springer-Verlag. IMA Vol. Math. Appl. 42, 89-97 (1992).

This paper discusses a variety of open problems involving the metric properties of harmonic measure (for the Laplacian) on domains in the plane. Most of the problems deal with the relation between harmonic measure and Hausdorff measures and particularly with 1-dimensional measure. There is also a discussion of recent related work by Jones, Makarov, Wolff and the author, among others. The paper ends by discussing some problems involving the geometry of Brownian paths in the plane and the random growth model called diffusion limited aggregation.
For the entire collection see [Zbl 0762.00004].

Reviewer: Ch.J.Bishop

MSC:

30C35	General theory of conformal mappings
30C55	General theory of univalent and multivalent functions of one complex variable

Keywords:

twist points; eigenvalues of Laplacian; harmonic measure