On the mixed problem for harmonic functions in a 2-D exterior cracked domain with Neumann condition on cracks

Author:
P. A. Krutitskii

Journal:
Quart. Appl. Math. **65** (2007), 25-42

MSC (2000):
Primary 35J05, 35J25

DOI:
https://doi.org/10.1090/S0033-569X-07-01046-1

Published electronically:
January 2, 2007

MathSciNet review:
2313147

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Abstract: The mixed Dirichlet-Neumann problem for the Laplace equation in an unbounded plane domain with cuts (cracks) is studied. The Dirichlet condition is given on closed curves making up the boundary of the domain, while the Neumann condition is specified on the cuts. The existence of a classical solution is proved by potential theory and a boundary integral equation method. The integral representation for a solution is obtained in the form of potentials. The density of the potentials satisfies a uniquely solvable Fredholm integral equation of the second kind and index zero. Singularities of the gradient of the solution at the tips of the cuts are investigated.

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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-07-01046-1

Keywords:
Laplace equation,
Dirichlet--Neumann problem,
boundary integral equation method.

Received by editor(s):
October 27, 2005

Published electronically:
January 2, 2007

Article copyright:
© Copyright 2007
Brown University