Mathematical Problems in Engineering 
Volume 7 (2001), Issue 1, Pages 1-13
doi:10.1155/S1024123X01001491

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

Abstract

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.