The Dirichlet Problem for the 2D Laplace Equation in a Domain with Cracks without Compatibility Conditions at Tips of the Cracks

We study the Dirichlet problem for the 2D Laplace equation in a domain bounded by smooth closed curves and smooth cracks. In the formulation of the problem, we do not require compatibility conditions for Dirichlet’s boundary data at the tips of the cracks. However, if boundary data satisfies the compatibility conditions at the tips of the cracks, then this is a particular case of our problem. The cases of both interior and exterior domains are considered. The wellposed 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 weak solution of the problem does not typically exist, though the classical solution exists. The asymptotic formulae for singularities of a solution gradient at the tips of the cracks are presented.