Abstract and Applied Analysis
Volume 2004 (2004), Issue 6, Pages 501-510
doi:10.1155/S1085337504306196

Which solutions of the third problem for the Poisson equation are bounded?

Dagmar Medková1,2

1Mathematical Institute, Academy of Sciences of the Czech Republic, Žitná 25, Praha 1 115 67, Czech Republic
2Department of Technical Mathematics, Faculty of Mechanical Engineering, Czech Technical University, Karlovo nám. 13, Praha 2 121 35, Czech Republic

Received 10 September 2002

Copyright © 2004 Dagmar Medková. 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

This paper deals with the problem Δu=g on G and ∂u/∂n+uf=L on ∂G. Here, G⊂ℝm, m>2, is a bounded domain with Lyapunov boundary, f is a bounded nonnegative function on the boundary of G, L is a bounded linear functional on W1,2(G) representable by a real measure μ on the boundary of G, and g∈L2(G)∩Lp(G), p>m/2. It is shown that a weak solution of this problem is bounded in G if and only if the Newtonian potential corresponding to the boundary condition μ is bounded in G.