Nicht-tangentiales Randverhalten von harmonischen Funktionen in irregulären Randpunkten. (Non-tangential boundary behaviour of harmonic functions in irregular boundary points) (Q1086392)

Let z be an irregular boundary point of a domain \(U\subset {\mathbb{R}}^ n\) and \(A\subset U\) be non-tangential at z (i.e. \(z\in \bar A\) and inf\(\{\) \(| x-y| /| x-z|:\) \(x\in A\), \(y\in {\mathbb{R}}^ n\setminus U\}>0)\). Then it is shown that \(\lim _{A\ni x\to z}h(z)\) exists for any positive harmonic function h on U. This strongly contrasts the non- tangential boundary behavior at regular boundary points. There one can only show almost everywhere non-tangential convergence under strong conditions on U. The above result can be generalized to harmonic spaces satisfying certain additional conditions which are fulfilled for all types of elliptic second order equations.