Mathematical Problems in Engineering
Volume 7 (2001), Issue 1, Pages 1-13

Explicit solution of the jump problem for the Laplace equation and singularities at the edges

P. A. Krutitskii

Department of Mathematics, Faculty of Physics, Moscow State University, Moscow 119899, Russia

Received 14 December 1999

Copyright © 2001 P. A. Krutitskii. 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 cuts in a plane is studied. The jump of the solution of the Laplace equation and the jump of its normal derivative are specified on the cuts. The problem is studied under different conditions at infinity, which lead to different uniqueness and existence theorems. The solution of this problem is constructed in the explicit form by means of single layer and angular potentials. The singularities at the ends of the cuts are investigated.