Fine behaviour of solutions of the Dirichlet problem near an irregular point. (English) Zbl 0699.31015

Netuka, Ivan

Bull. Sci. Math., II. Sér. 114, No. 1, 1-22 (1990).

The behaviour of the generalized solution \(H^ uf\) of the Dirichlet problem, on a bounded open set u with resolutive boundary data f, is investigated near an irregular boundary point z. Since z is irregular, \(H^ uf(y)\) may not have a limit as y approaches z. However, if f is lower bounded, then \(H^ uf\) has a fine limit and the author shows that this limit is equal to the integral of f with respect to the balayage of the Dirac measure.

Reviewer: P.M.Gauthier

MSC:

31B25	Boundary behavior of harmonic functions in higher dimensions
31D05	Axiomatic potential theory
31B35	Connections of harmonic functions with differential equations in higher dimensions

Keywords:

Dirichlet problem; irregular boundary point; fine limit; balayage