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## Charles L. Epstein and Rafe Mazzeo

Print publication date: 2013

Print ISBN-13: 9780691157122

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691157122.001.0001

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# Maximum Principles and Uniqueness Theorems

Chapter:
(p.34) Chapter Three Maximum Principles and Uniqueness Theorems
Source:
Degenerate Diffusion Operators Arising in Population Biology (AM-185)
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691157122.003.0003

This chapter proves maximum principles for two parabolic and elliptic equations from which the uniqueness results follow easily. It also considers the main consequences of the maximum principle, both for the model operators on an open orthant and for the general Kimura diffusion operators on a compact manifold with corners, as well as their elliptic analogues. Of particular note in this regard is a generalization of the Hopf boundary point maximum principle. The chapter first presents maximum principles for the model operators before discussing Kimura diffusion operators on manifolds with corners. It then describes maximum principles for the heat equation as well as the corresponding maximum principle and uniqueness result for Kimura diffusion equations.

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