Abstract
In this paper, we develop a new approach to establish gradient estimates for positive solutions to the heat equation of elliptic or subelliptic operators on Euclidean spaces or on Riemannian manifolds.More precisely, we give some estimatesof the gradient of logarithm of a positive solution via the uniform bound ofthe logarithm of the solution. Moreover, we give a generalized version ofLi-Yau's estimate. Our proof is based on the link between PDE and quadraticBSDE. Our method might be useful to study some (nonlinear) PDEs.
Highlights
We study positive solutions u of a linear parabolic equation
In this paper we prove several gradient estimates for the positive solutions of (1.1)
Let us begin with an interesting result about BSDEs with quadratic growth
Summary
Where M is either the Euclidean space Rn and L is an elliptic or sub-elliptic operator of secondorder. In the same paper [20], Li and Yau obtained a gradient estimate for positive solutions in terms of the dimension and a lower bound (which may be negative) of the Ricci curvature, though less precise. Their estimates in negative case have been improved over the years, see for example [27], [28], [2] and [1]. Theorem 1.4 Let M be a complete manifold of dimension n with non-negative Ricci curvature. Last section is devoted to establish a generalized Li-Yau estimate via analytic tool
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
