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 |
|---|---|
| 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 1 |
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All Science Journal Classification (ASJC) codes
- Analysis
- Applied Mathematics
Cite this
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Harnack inequality for nondivergent parabolic operators on Riemannian manifolds. / Kim, Seick; Kim, Soojung; Lee, Ki Ahm.
In: Calculus of Variations and Partial Differential Equations, Vol. 49, No. 1-2, 01.01.2014, p. 669-706.Research output: Contribution to journal › Article
TY - JOUR
T1 - Harnack inequality for nondivergent parabolic operators on Riemannian manifolds
AU - Kim, Seick
AU - Kim, Soojung
AU - Lee, Ki Ahm
PY - 2014/1/1
Y1 - 2014/1/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=84891902215&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84891902215&partnerID=8YFLogxK
U2 - 10.1007/s00526-013-0596-6
DO - 10.1007/s00526-013-0596-6
M3 - Article
AN - SCOPUS:84891902215
VL - 49
SP - 669
EP - 706
JO - Calculus of Variations and Partial Differential Equations
JF - Calculus of Variations and Partial Differential Equations
SN - 0944-2669
IS - 1-2
ER -