On existence of a classical solution and non-existence of a weak solution to the Dirichlet problem in a planar domain with slits for Laplacian
П. А. Крутицкий · Quarterly of Applied Mathematics · 2007
The Dirichlet problem for the Laplacian in a planar domain bounded by smooth closed curves and smooth double-sided open arcs (slits) is considered in the case when the solution is not continuous at the ends of the slits. The cases of both interior and exterior domains are considered. The well-posed formulation of the problem is given, theorems on existence and uniqueness of a classical solution are proved, and the integral representation for a solution is obtained. It is shown that a weak solution of the problem does not typically exist, though the classical solution exists.