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International Journal of Mathematics and Mathematical Sciences
Volume 2006, Article ID 91983, 14 pages
http://dx.doi.org/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

Dagmar Medková

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 Hindawi Publishing Corporation. 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.