Sharp modulus of continuity for parabolic equations on manifolds and lower bounds for the first eigenvalue

Ben Andrews; Julie Clutterbuck

doi:10.2140/apde.2013.6.1013

Sharp modulus of continuity for parabolic equations on manifolds and lower bounds for the first eigenvalue

Ben Andrews^*, Julie Clutterbuck

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

35 Citations (Scopus)

Abstract

We derive sharp estimates on the modulus of continuity for solutions of the heat equation on a compact Riemannian manifold with a Ricci curvature bound, in terms of initial oscillation and elapsed time. As an application, we give an easy proof of the optimal lower bound on the first eigenvalue of the Laplacian on such a manifold as a function of diameter.

Original language	English
Pages (from-to)	1013-1024
Number of pages	12
Journal	Analysis and PDE
Volume	6
Issue number	5
DOIs	https://doi.org/10.2140/apde.2013.6.1013
Publication status	Published - 2013

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abstract = "We derive sharp estimates on the modulus of continuity for solutions of the heat equation on a compact Riemannian manifold with a Ricci curvature bound, in terms of initial oscillation and elapsed time. As an application, we give an easy proof of the optimal lower bound on the first eigenvalue of the Laplacian on such a manifold as a function of diameter.",

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Sharp modulus of continuity for parabolic equations on manifolds and lower bounds for the first eigenvalue

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