BOUNDARY-VALUE-PROBLEMS AND BROWNIAN-MOTION ON FRACTALS

A physical state in a domain is often described by a model containing a linear partial differential equation and associated boundary conditi ons. The mathematical tools required to study this are well known if t he boundary of the domain is smooth enough or if the boundary is smoot h except for one or several corners. But in reality the boundary of th e domain is usually not smooth. The typical situation is rather that t he boundary is strongly broken with an intricate detailed structure an d maybe that the boundary exhibits similar patterns in different scale s. This means that the boundary is typically a fractal showing some ki nd of self-similarity: a magnification of a part of the boundary has, in some sense, the same structure as the whole boundary. A typical exa mple of a domain in the plane having a boundary of this kind is von Ko ch's snowflake domain. rn the case of a fractal boundary the classical tools and theorems no longer hold. How does one provide the mathemati cal background in this case? This is the main topic of this survey pap er. However, we also study Brownian motion on fractals. (C) 1997 Elsev ier Science Ltd All rights reserved.