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 language | English |
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Pages (from-to) | 669-706 |
Number of pages | 38 |
Journal | Calculus of Variations and Partial Differential Equations |
Volume | 49 |
Issue number | 1-2 |
DOIs | |
Publication status | Published - 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