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 conditions. The mathematical tools required to study this are well known if the boundary of the domain is smooth enough or if the boundary is smooth except for one or several corners. But in reality the boundary of the domain is usually not smooth. The typical situation is rather that the boundary is strongly broken with an intricate detailed structure and maybe that the boundary exhibits similar patterns in different scales. This means that the boundary is typically a fractal showing some kind of self-similarity: a magnification of a part of the boundary has, in some sense, the same structure as the whole boundary. A typical example of a domain in the plane having a boundary of this kind is von Kochʹs snowflake domain. In the case of a fractal boundary the classical tools and theorems no longer hold. How does one provide the mathematical background in this case? This is the main topic of this survey paper. However, we also study Brownian motion on fractals.