Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-24T05:59:03.678Z Has data issue: false hasContentIssue false
  • English
  • Français

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, [email protected] 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, [email protected]
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)