Longman-Pitmann Research Notes
in Mathematics Series 360

C.G. Simader and H. Sohr

The Dirichlet problem for the
Laplacian in bounded and
unbounded domains


1996. 294 pages
ISBN 0 582 20953 6
The Dirichlet problem for the Laplacian is one of the basic problems in the theory of partial differential equations. It plays a fundamental role in mathematical physics and engineering. In most cases bounded domains have been considered. A classical functional analytical approach via the Riesz representation theorem in Sobolev spaces leads to weak solutions not yet having the desired differentiability properties. Such properties can be shown in a subsequent step if some smoothness properties are satisfied. The purpose of the underlying Research Note is to extend these results to unbounded domains. Mainly the case of exterior domians is studied using appropriate generalized Sobolev-type spaces. The procedure uses many properties of harmonic polynomials. The Research Note is mainly directed to researchers and graduate students working in mathematics, physics and engineering scienes.
Letzte Änderung: 17.12.2001