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metric measure manifolds

Proceedings of the London Mathematical Society (1)

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EIGENVALUES AND EIGENFUNCTIONS OF METRIC MEASURE MANIFOLDS
YUXIN GE, ZHONGMIN SHEN
Journal:

Proceedings of the London Mathematical Society / Volume 82 / Issue 3 / May 2000

Published online by Cambridge University Press:

20 August 2001, pp. 725-746

Print publication:

May 2000
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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.