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.

Abstract

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.