Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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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 | References | Similar Articles | Additional Information

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

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