Harnack inequality for nondivergent parabolic operators on Riemannian manifolds

Seick Kim, Soojung Kim, Ki Ahm Lee

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

We consider second-order linear parabolic operators in non-divergence form that are intrinsically defined on Riemannian manifolds. In the elliptic case, Cabré proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional curvature of the underlying manifold is nonnegative. Later, Kim improved Cabré's result by replacing the curvature condition by a certain condition on the distance function. Assuming essentially the same condition introduced by Kim, we establish Krylov-Safonov Harnack inequality for nonnegative solutions of the non-divergent parabolic equation. This, in particular, gives a new proof for Li-Yau Harnack inequality for positive solutions to the heat equation in a manifold with nonnegative Ricci curvature.

Original languageEnglish
Pages (from-to)669-706
Number of pages38
JournalCalculus of Variations and Partial Differential Equations
Volume49
Issue number1-2
DOIs
Publication statusPublished - 2014 Jan 1

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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