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Some results and examples about the behavior of harmonic functions and Green’s functions with respect to second order elliptic operators

Published online by Cambridge University Press:  22 January 2016

Alano Ancona
Alano Ancona*
Affiliation:
Département de Mathématiques, Bâtiment 425, Université Paris-Sud, 91405 Orsay, France, ancona@matups.math.u-psud.fr
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Abstract

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Let M be a manifold and let L be a sufficiently smooth second order elliptic operator in M such that (M, L) is a transient pair. It is first shown that if L is symmetric with respect to some density in M, there exists a positive L-harmonic function in M which dominates L-Green’s function at infinity. Other classes of elliptic operators are investigated and examples are constructed showing that this property may fail if the symmetry assumption is removed. Another part of the paper deals with the existence of critical points for certain L-harmonic functions with periodicity properties. A class of small perturbations of second order elliptic operators is also described.

Keywords

31C1231C3560J1060J60
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 165 , March 2002 , pp. 123 - 158
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2002

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