International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 91983, 14 pages
doi:10.1155/IJMMS/2006/91983
The solution of the third problem for the Laplace equation on
planar domains with smooth boundary and inside cracks and modified
jump conditions on cracks
Department of Technical Mathematics, Faculty of Mechanical Engineering, Czech Technical University, Karlovo Nám. 13, 12135 Praha 2, Czech Republic
Received 23 November 2005; Revised 13 March 2006; Accepted 4 April 2006
Copyright © 2006 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 studies the third problem for the Laplace equation on a
bounded planar domain with inside cracks. The third condition
∂u/∂n+hu=f
is given on the boundary of the
domain. The skip of the function u+−u−=g
and the modified
skip of the normal derivatives (∂u/∂n)+−(∂u/∂n)−+hu+=f
are given on cracks. The
solution is looked for in the form of the sum of a modified
single-layer potential and a double-layer potential. The solution
of the corresponding integral equation is constructed.