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

Author:
P. A. Krutitskii

Journal:
Quart. Appl. Math. **67** (2009), 379-399

MSC (2000):
Primary 35J05, 35J25

DOI:
https://doi.org/10.1090/S0033-569X-09-01130-1

Published electronically:
March 26, 2009

MathSciNet review:
2514640

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Abstract: The Dirichlet problem for the Laplacian in a planar multiply connected interior domain bounded by smooth closed curves is considered in the case when the boundary data is piecewise continuous; i.e., it may have jumps in certain points of the boundary. It is assumed that the solution to the problem may not be continuous at the same points. 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 classical solution is obtained. The problem is reduced to a uniquely solvable Fredholm integral equation of the second kind and of index zero. It is shown that a weak solution to the problem does not exist typically, though the classical solution exists.

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Additional Information

**P. A. Krutitskii**

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

DOI:
https://doi.org/10.1090/S0033-569X-09-01130-1

Keywords:
Laplace equation,
Dirichlet problem,
discontinuous boundary data.

Received by editor(s):
February 14, 2008

Published electronically:
March 26, 2009

Article copyright:
© Copyright 2009
Brown University