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Elliptic PDEs on Compact Ricci Limit Spaces and Applications
About this Title
Shouhei Honda, Faculty of Mathmatics, Kyushu University, 744, Motooka, Nishi-Ku, Fukuoka 819-0395, Japan
Publication: Memoirs of the American Mathematical Society
Publication Year:
2018; Volume 253, Number 1211
ISBNs: 978-1-4704-2854-9 (print); 978-1-4704-4417-4 (online)
DOI: https://doi.org/10.1090/memo/1211
Published electronically: March 7, 2018
Keywords: Gromov-Hausdorff convergence,
Ricci curvature,
Elliptic PDEs.
MSC: Primary 53C20
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries
- 3. $L^p$-convergence revisited
- 4. Poisson’s equations
- 5. Schrödinger operators and generalized Yamabe constants
- 6. Rellich type compactness for tensor fields
- 7. Differential forms
Abstract
In this paper we study elliptic PDEs on compact Gromov-Hausdorff limit spaces of Riemannian manifolds with lower Ricci curvature bounds. In particular we establish continuities of geometric quantities, which include solutions of Poisson’s equations, eigenvalues of Schrödinger operators, generalized Yamabe constants and eigenvalues of the Hodge Laplacian, with respect to the Gromov-Hausdorff topology. We apply these to the study of second-order differential calculus on such limit spaces.- Kazuo Akutagawa, Gilles Carron, and Rafe Mazzeo, The Yamabe problem on stratified spaces, Geom. Funct. Anal. 24 (2014), no. 4, 1039–1079. MR 3248479, DOI 10.1007/s00039-014-0298-z
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