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EIGENVALUES AND EIGENFUNCTIONS OF METRIC MEASURE MANIFOLDS

Published online by Cambridge University Press:  20 August 2001

YUXIN GE  and
ZHONGMIN SHEN
YUXIN GE
Affiliation:
C.M.L.A., E.N.S de Cachan, 61 avenue du Président Wilson, 94235 Cachan Cedex, Francege@cmla.ens-cachan.fr and Département de Mathématiques, Faculté de Sciences et Technologie, Université Paris XII-Val de Marne, 61 avenue du Général de Gaulle, 94010 Crétail Cedex, France
ZHONGMIN SHEN
Affiliation:
Department of Mathematical Sciences, IUPUI, Indianapolis, IN 46202-3216, USAzshen@math.iupui.edu
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Abstract

In this paper we study the eigenvalues and eigenfunctions of metric measure manifolds. We prove that any eigenfunction is $C^{1,\alpha}$ at its critical points and $C^{\infty}$ elsewhere. Moreover, the eigenfunction corresponding to the first eigenvalue in the Dirichlet problem does not change sign. We also discuss the first eigenvalue, the Sobolev constants and their relationship with the isoperimetric constants. 2000 Mathematics Subject Classification: 47J05, 47J10, 53C60, 58E05, 58C40.

Keywords

metric measure manifoldsFinsler metricvolume formLaplacianeigenvalueeigenfunctionisoperimetric constantSobolev constant
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 82 , Issue 3 , May 2000 , pp. 725 - 746
Copyright
2001 London Mathematical Society

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