Harnack inequality for nondivergent parabolic operators on Riemannian manifolds

Seick Kim, Soojung Kim, Ki Ahm Lee

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)


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
Issue number1-2
Publication statusPublished - 2014 Jan

Bibliographical note

Funding Information:
Seick Kim is supported by NRF Grant No. 2012-040411 and TJ Park Junior Faculty Fellowship. Ki-Ahm Lee was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2008-314-C00023).

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics


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