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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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Existence of Solutions

Existence of Solutions

Chapter:
(p.181) Chapter Ten Existence of Solutions
Source:
Degenerate Diffusion Operators Arising in Population Biology (AM-185)
Author(s):

Charles L. Epstein

Rafe1 Mazzeo

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691157122.003.0010

This chapter proves existence of solutions to the inhomogeneous problem using the Schauder estimate and analyzes a generalized Kimura diffusion operator, L, defined on a manifold with corners, P. The discussion centers on the solution w = v + u, where v solves the homogeneous Cauchy problem with v(x, 0) = f(x) and u solves the inhomogeneous problem with u(x, 0) = 0. The chapter first provides definitions for the Wright–Fisher–Hölder spaces on a general compact manifold with corners before explaining the steps involved in the existence proof. It then verifies the induction hypothesis and treats the k = 0 case. It also shows how to perform the doubling construction for P and considers the existence of the resolvent operator and a contraction semi-group. Finally, it discusses the problem of higher regularity.

Keywords:   inhomogeneous problem, generalized Kimura diffusion, manifold with corners, homogeneous Cauchy problem, Hölder space, induction hypothesis, doubling, resolvent operator, semi-group, higher regularity

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