EXISTENCE OF A CLASSICAL SOLUTION AND NON–EXISTENCE OF A WEAK SOLUTION TO THE HARMONIC DIRICHLET PROBLEM IN A PLANAR DOMAIN WITH CRACKS

The harmonic Dirichlet problem in a planar domain with smooth cracks of an arbitrary shape is considered in case, when the solution is not continuous at the ends the cracks. 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 solution is obtained. With the help of the integral representation, the properties of the solution are studied. It is proved that a weak solution of the Dirichlet problem in question does not exist typically, though the classical solution exists.