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.
|Number of pages||38|
|Journal||Calculus of Variations and Partial Differential Equations|
|Publication status||Published - 2014 Jan|
Bibliographical noteFunding 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
- Applied Mathematics