International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 36, Pages 1909-1921
doi:10.1155/S0161171204308094
The Poisson equation in homogeneous Sobolev spaces
Fachbereich 17 Mathematik/Informatik, Universität Kassel, Heinrich-Plett-Str. 40, Kassel 34109, Germany
Received 9 August 2003
Copyright © 2004 Tatiana Samrowski and Werner Varnhorn. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We consider Poisson's equation in an n-dimensional exterior
domain G(n≥2) with a sufficiently smooth boundary. We
prove that for external forces and boundary values given in
certain Lq(G)-spaces there exists a solution in the
homogeneous Sobolev space S2,q(G), containing functions
being local in Lq(G) and having second-order derivatives in
Lq(G) Concerning the uniqueness of this solution we prove
that the corresponding nullspace has the dimension n+1, independent of q.