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International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 579457, 12 pages
Research Article

On the Modified Jump Problem for the Laplace Equation in the Exterior of Cracks in a Plane

1KIAM, Miusskaya Sq. 4, Moscow 125047, Russia
2AIST, 1-2-1 Namiki, Tsukuba, Ibaraki 305-8564, Japan

Received 20 March 2012; Revised 29 March 2012; Accepted 14 April 2012

Academic Editor: Vladimir Mityushev

Copyright © 2012 P. A. Krutitskii and A. Sasamoto. 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.


The boundary value problem for the Laplace equation outside several cracks in a plane is studied. The jump of the solution of the Laplace equation and the boundary condition containing the jump of its normal derivative are specified on the cracks. The problem has unique classical solution under certain conditions. The new integral representation for the unique solution of this problem is obtained. The problem is reduced to the uniquely solvable Fredholm equation of the second kind and index zero. The integral representation and integral equation are essentially simpler than those derived for this problem earlier. The singularities at the ends of the cracks are investigated.