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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Green's function, harmonic transplantation,
and best Sobolev constant
in spaces of constant curvature


Authors: C. Bandle, A. Brillard and M. Flucher
Journal: Trans. Amer. Math. Soc. 350 (1998), 1103-1128
MSC (1991): Primary 49S05, 35J25, 35J65
MathSciNet review: 1458294
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Abstract | References | Similar Articles | Additional Information

Abstract: We extend the method of harmonic transplantation from Euclidean domains to spaces of constant positive or negative curvature. To this end the structure of the Green's function of the corresponding Laplace-Beltrami operator is investigated. By means of isoperimetric inequalities we derive complementary estimates for its distribution function. We apply the method of harmonic transplantation to the question of whether the best Sobolev constant for the critical exponent is attained, i.e. whether there is an extremal function for the best Sobolev constant in spaces of constant curvature. A fairly complete answer is given, based on a concentration-compactness argument and a Pohozaev identity. The result depends on the curvature.


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Additional Information

C. Bandle
Affiliation: Universität Basel, Mathematisches Institut, Rheinsprung 21, CH–4051 Basel, Schweiz
Email: bandle@math.unibas.ch

A. Brillard
Affiliation: Université de Haute Alsace, Faculté des Sciences Techniques, 4 rue des Frères Lumière, F–68093 Mulhouse Cédex, France
Email: A.Brillard@univ--mulhouse.fr

M. Flucher
Affiliation: Universität Basel, Mathematisches Institut, Rheinsprung 21, CH–4051 Basel, Schweiz
Email: flucher@math.unibas.ch

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02085-6
PII: S 0002-9947(98)02085-6
Keywords: Green's function, harmonic transplantation, best Sobolev constant
Received by editor(s): June 15, 1996
Article copyright: © Copyright 1998 American Mathematical Society