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Neumann problem on a half-space
Authors:
Fumiyama Shu, Masaki Tanaka and Minoru Yanagishita
Journal:
Proc. Amer. Math. Soc. 139 (2011), 1333-1345
MSC (2000):
Primary 31B05; Secondary 31B20
Posted:
November 30, 2010
MathSciNet review:
2748426
Full-text PDF
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Additional Information
Abstract: In this paper, a solution of the Neumann problem on a half-space for a slowly growing continuous boundary function is constructed by the generalized Neumann integral with this boundary function. The relation between this particular solution and certain general solutions is discussed. A solution of the Neumann problem for any continuous boundary function is also given explicitly by the Neumann integral with the generalized Neumann kernel depending on this boundary function.
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- D.H. Armitage, The Neumann problem for a function harmonic in
 , Arch. Rational Mech. Anal., 63 (1976), 89-105. MR 0427656 (55:687)
- 2.
- -, On harmonic polynomials, Proc. London Math. Soc., (3) 34 (1977), 173-192. MR 0427661 (55:692)
- 3.
- -, Representations of harmonic functions in half-spaces, Proc. London Math. Soc., (3) 38 (1979), 53-71. MR 520972 (80d:31003)
- 4.
- D.H. Armitage and S.J. Gardiner, Classical Potential Theory, Springer-Verlag London Ltd., 2001. MR 1801253 (2001m:31001)
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- M. Brelot, Éléments de la théorie classique du potentiel, Centre de Documentation Universitaire, Paris, 1965. MR 0178154 (31:2412)
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- M. Finkelstein and S. Scheinberg, Kernels for solving problems of Dirichlet type in a half-plane, Advances in Math., 18 (1975), 108-113. MR 0382677 (52:3559)
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- S.J. Gardiner, The Dirichlet and Neumann problems for harmonic functions in half-spaces, J. London Math. Soc., (2) 24 (1981), 502-512. MR 635881 (82m:31006)
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- C. Müller, Spherical harmonics, Springer-Verlag, Berlin, 1966. MR 0199449 (33:7593)
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Additional Information
Fumiyama Shu
Affiliation:
Rakuten, Inc., 604, 1778-1 Hisasue, Takatsu-ku, Kawasaki-shi, Kanagawa 213-0026, Japan
Email:
fumiyama.shu@mail.rakuten.com.tw
Masaki Tanaka
Affiliation:
Department of Mathematics and Informatics, Division of Fundamental Science, Graduate School of Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan
Email:
emblem-yp5@graduate.chiba-u.jp
Minoru Yanagishita
Affiliation:
Department of Mathematics and Informatics, Division of Fundamental Science, Graduate School of Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan
Email:
myanagis@g.math.s.chiba-u.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002-9939-2010-10787-6
PII:
S 0002-9939(2010)10787-6
Keywords:
Half-space,
Neumann problem,
Neumann integral
Received by editor(s):
November 20, 2009
Posted:
November 30, 2010
Communicated by:
Mario Bonk
Article copyright:
© Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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