### 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 1 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

*Calculus of Variations and Partial Differential Equations*,

*49*(1-2), 669-706. https://doi.org/10.1007/s00526-013-0596-6

}

*Calculus of Variations and Partial Differential Equations*, vol. 49, no. 1-2, pp. 669-706. https://doi.org/10.1007/s00526-013-0596-6

**Harnack inequality for nondivergent parabolic operators on Riemannian manifolds.** / Kim, Seick; Kim, Soojung; Lee, Ki Ahm.

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

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 -