On existence of a classical solution and non-existence of a weak solution to the Dirichlet problem in a planar domain with slits for Laplacian

Author:
P. A. Krutitskii

Journal:
Quart. Appl. Math. **66** (2008), 177-190

MSC (2000):
Primary 35J05, 35J25

DOI:
https://doi.org/10.1090/S0033-569X-07-01067-3

Published electronically:
December 7, 2007

MathSciNet review:
2396656

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Abstract: The Dirichlet problem for the Laplacian in a planar domain bounded by smooth closed curves and smooth double-sided open arcs (slits) is considered in the case when the solution is not continuous at the ends of the slits. 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 a weak solution of the problem does not typically exist, though the classical solution exists.

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Additional Information

**P. A. Krutitskii**

Affiliation:
KIAM, Department 25, Miusskaya Sq. 4, Moscow 125047, Russia

DOI:
https://doi.org/10.1090/S0033-569X-07-01067-3

Received by editor(s):
April 6, 2007

Published electronically:
December 7, 2007

Article copyright:
© Copyright 2007
Brown University