Martin boundary of a fractal domain

Journal article, 2003

A uniformly John domain is a domain intermediate between a John domain and a uniform domain. We determine the Martin boundary of a uniformly John domain D as an application of a boundary Harnack principle. We show that a certain self-similar fractal has its complement as a uniformly John domain. In particular, the complement of the 3-dimensional Sierpinacuteski gasket is a uniform domain and its Martin boundary is homeomorphic to the Sierpinacuteski gasket itself.