On the properties of solutions of the harmonic Dirichlet problem in a two-dimensional domain with cuts
П. А. Крутицкий · Differential Equations · 2010
We consider the Dirichlet problem for the Laplace equation in a plane domain with smooth cuts of arbitrary form for the case in which the solution is not continuous at the endpoints of the cuts. We present a well-posed statement of the problem, prove the existence and uniqueness theorems for the classical solution, obtain an integral representation of the solution, and use it to analyze the properties of the solution. We show that, as a rule, the Dirichlet problem in this setting has no weak solutions, even though there exists a classical solution.