The Dirichlet problem for the Laplacian with discontinuous boundary data in a 2D multiply connected exterior domain

Abstract

The Dirichlet problem for Laplacian in a planar multiply connected exterior domain bounded by smooth closed curves is considered in case, when the boundary data is piecewise continuous, i.e. it may have jumps in certain points of the boundary. It is assumed that the solution to the problem may be not continuous at the same points. The well-posed formulation of the problem is given, theorems on existence and uniqueness of a classical solution are proved, the integral representation for a classical solution is obtained. The problem is reduced to a uniquely solvable Fredholm integral equation of the second kind and of index zero. It is shown that a weak solution to the problem does not exist typically, though the classical solution exists.