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Degenerate Diffusion Operators Arising in Population Biology (AM-185)$
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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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Holder Estimates for General Models

Holder Estimates for General Models

(p.143) Chapter Nine Hölder Estimates for General Models
Degenerate Diffusion Operators Arising in Population Biology (AM-185)

Charles L. Epstein

Rafe1 Mazzeo

Princeton University Press

This chapter presents the Hölder estimates for general model problems. It first estimates solutions to heat equations for both the homogeneous Cauchy problem and the inhomogeneous problem, obtaining first and second derivative estimates in the latter case, before discussing a general result describing the off-diagonal and long-time behavior of the solution kernel for the general model. It also states a proposition summarizing the properties of the resolvent operator as an operator on the Hölder spaces. In contrast to the case of the heat equation, there is no need to assume that the data has compact support in the x-variables to prove estimates when k > 0.

Keywords:   general model problem, heat equation, homogeneous Cauchy problem, inhomogeneous problem, off-diagonal behavior, long-time behavior, resolvent operator, Hölder space

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