Green’s function, harmonic transplantation, and best Sobolev constant in spaces of constant curvature
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- by C. Bandle, A. Brillard and M. Flucher PDF
- Trans. Amer. Math. Soc. 350 (1998), 1103-1128 Request permission
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.References
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Additional Information
- C. Bandle
- Affiliation: Universität Basel, Mathematisches Institut, Rheinsprung 21, CH–4051 Basel, Schweiz
- MR Author ID: 30425
- 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
- Received by editor(s): June 15, 1996
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 350 (1998), 1103-1128
- MSC (1991): Primary 49S05, 35J25, 35J65
- DOI: https://doi.org/10.1090/S0002-9947-98-02085-6
- MathSciNet review: 1458294