scholarly journals On existence of a classical solution and nonexistence of a weak solution to the Dirichlet problem for the Laplacian with discontinuous boundary data

2009 ◽  
Vol 67 (2) ◽  
pp. 379-399
Author(s):  
P. A. Krutitskii
Keyword(s):  
Weak Solution ◽  
Dirichlet Problem ◽  
Classical Solution ◽  
Boundary Data ◽  
Discontinuous Boundary ◽  
Discontinuous Boundary Data

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

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
P. A. Krutitskii
Keyword(s):  
Dirichlet Problem ◽  
Classical Solution ◽  
Laplace Equation ◽  
Boundary Data ◽  
Exterior Domains ◽  
Compatibility Conditions ◽  
Closed Curves ◽  
Asymptotic Formulae ◽  
Well Posed ◽  
Solution Gradient

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 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 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.

Sign in / Sign up

Export Citation Format

Share Document