Existence of a Classical Solution and Nonexistence of a Weak Solution to the Dirichlet Problem for the Laplace Equation in a Plane Domain with Cracks

Plane domains with cracks are plane domains bounded by closed curves and open arcs (cracks). Boundary value problems in such domains model cracked solid bodies or obstacles and screens (or wings) in fluids. An integral representation of a classical solution to the harmonic Dirichlet problem in a plane domain with cracks of an arbitrary shape has been obtained by the method of integral equations in [Kr00-1], [Kr00-2], [Kr98], [Kr97], [Kr05] in the case when the solution is assumed to be continuous at the ends of the cracks. In this chapter this problem is considered in the case when the solution is not continuous at the ends of the cracks. The well-posed formulation of the boundary value problem is given, theorems on existence and uniqueness of a classical solution are proved, and the integral representation for a classical solution is obtained. Moreover, properties of the solution are studied with the help of this integral representation. It appears that the classical solution to the Dirichlet problem considered in this chapter exists, while the weak solution typically does not exist, though both the cracks and the functions specified in the boundary conditions are smooth enough. This result follows from the fact that the square of the gradient of a classical solution basically is not integrable near the ends of the cracks, since singularities of the gradient are rather strong there. This result is very important for numerical analysis; it shows that finite elements and finite difference methods cannot be applied to numerical treatment of the Dirichlet problem in question directly, since all these methods imply existence of a weak solution. To use difference methods for numerical analysis, one has to localize all strong singularities first and next use a difference method in a domain excluding the neighborhoods of the singularities.