Fractality and the Internal Dirichlet Problem

Abstract

In this chapter influence of fractality on solution of the two-dimensional internal Dirichlet problem is analyzed. Two different situations are considered namely the first of them deals with fractal boundary condition on the unit disk. In this case exact solution of the Laplace equation proves to obey to some analog of the de Rham functional equation. Also norm and the Dirichlet integral for this solution has been estimated. In the second situation boundary condition is supposed to be regular but boundary of the domain is fractally perturbed. For clarification of this case both approximate conformal mapping technique and the Potapov concept of physical fractals has been applied.