Harnack inequality for nondivergent parabolic operators on Riemannian manifolds

Seick Kim, Soojung Kim, Ki Ahm Lee

Research output: Contribution to journalArticle

6 Citations (Scopus)

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 languageEnglish
Pages (from-to)669-706
Number of pages38
JournalCalculus of Variations and Partial Differential Equations
Volume49
Issue number1-2
DOIs
Publication statusPublished - 2014 Jan 1

Fingerprint

Parabolic Operator
Harnack Inequality
Riemannian Manifold
Nonnegative Curvature
Nonnegative Solution
Ricci Curvature
Sectional Curvature
Distance Function
Heat Equation
Parabolic Equation
Linear Operator
Positive Solution
Curvature
Non-negative
Hot Temperature

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

@article{8a8810b5e6474c8ca012f3370a0645fd,
title = "Harnack inequality for nondivergent parabolic operators on Riemannian manifolds",
abstract = "We consider second-order linear parabolic operators in non-divergence form that are intrinsically defined on Riemannian manifolds. In the elliptic case, Cabr{\'e} proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional curvature of the underlying manifold is nonnegative. Later, Kim improved Cabr{\'e}'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.",
author = "Seick Kim and Soojung Kim and Lee, {Ki Ahm}",
year = "2014",
month = "1",
day = "1",
doi = "10.1007/s00526-013-0596-6",
language = "English",
volume = "49",
pages = "669--706",
journal = "Calculus of Variations and Partial Differential Equations",
issn = "0944-2669",
publisher = "Springer New York",
number = "1-2",

}

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 journalArticle

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 -