scholarly journals Gradient estimates for heat kernels and harmonic functions

2020 ◽  
Vol 278 (8) ◽  
pp. 108398 ◽  
Author(s):  
Thierry Coulhon ◽  
Renjin Jiang ◽  
Pekka Koskela ◽  
Adam Sikora
Keyword(s):  
Harmonic Functions ◽  
Heat Kernels ◽  
Gradient Estimates

Poisson’s integral formula via heat kernels

Analysis ◽  
2019 ◽  
Vol 39 (2) ◽  
pp. 59-64
Author(s):  
Yoichi Miyazaki
Keyword(s):  
Fourier Transform ◽  
Heat Kernel ◽  
Half Space ◽  
Unbounded Domain ◽  
Harmonic Functions ◽  
Integral Formula ◽  
Heat Kernels ◽  
Transform Method ◽  
Fourier Transform Method ◽  
The Fourier Transform

Abstract We give another proof of Poisson’s integral formula for harmonic functions in a ball or a half space by using heat kernels with Green’s formula. We wish to emphasize that this method works well even for a half space, which is an unbounded domain; the functions involved are integrable, since the heat kernel decays rapidly. This method needs no trick such as the subordination identity, which is indispensable when applying the Fourier transform method for a half space.

scholarly journals Uniform Gaussian Bounds for Subelliptic Heat Kernels and an Application to the Total Variation Flow of Graphs over Carnot Groups

2013 ◽  
Vol 1 ◽  
pp. 255-275 ◽  
Author(s):  
Luca Capogna ◽  
Giovanna Citti ◽  
Maria Manfredini
Keyword(s):  
Total Variation ◽  
Carnot Group ◽  
Riemannian Metrics ◽  
Heat Kernels ◽  
Gradient Estimates ◽  
Long Time Existence ◽  
Total Variation Flow ◽  
Long Time ◽  
Invariant Riemannian Metrics ◽  
Time Existence

Abstract In this paper we study heat kernels associated with a Carnot group G, endowed with a family of collapsing left-invariant Riemannian metrics σε which converge in the Gromov- Hausdorff sense to a sub-Riemannian structure on G as ε→ 0. The main new contribution are Gaussian-type bounds on the heat kernel for the σε metrics which are stable as ε→0 and extend the previous time-independent estimates in [16]. As an application we study well posedness of the total variation flow of graph surfaces over a bounded domain in a step two Carnot group (G; σε ). We establish interior and boundary gradient estimates, and develop a Schauder theory which are stable as ε → 0. As a consequence we obtain long time existence of smooth solutions of the sub-Riemannian flow (ε = 0), which in turn yield sub-Riemannian minimal surfaces as t → ∞.

Gradient estimates and heat kernel estimates

1995 ◽  
Vol 125 (5) ◽  
pp. 975-990 ◽  
Author(s):  
Zhongmin Qian
Keyword(s):  
Riemannian Manifold ◽  
Differential Operator ◽  
Heat Kernel ◽  
Harmonic Functions ◽  
Complete Riemannian Manifold ◽  
Gradient Estimates ◽  
Order Differential Operator ◽  
Kernel Estimates ◽  
Heat Kernel Estimates ◽  
Girsanov’S Formula

In the first part of this paper, Yau's estimates for positive L-harmonic functions and Li and Yau's gradient estimates for the positive solutions of a general parabolic heat equation on a complete Riemannian manifold are obtained by the use of Bakry and Emery's theory. In the second part we establish a heat kernel bound for a second-order differential operator which has a bounded and measurable drift, using Girsanov's formula.

Gradient estimates for p(x)-harmonic functions

2009 ◽  
Vol 131 (3-4) ◽  
pp. 403-414 ◽  
Author(s):  
S. Challal ◽  
A. Lyaghfouri
Keyword(s):  
Harmonic Functions ◽  
Gradient Estimates

Gradient Estimates for Harmonic Functions on Manifolds With Lipschitz Metrics

1998 ◽  
Vol 50 (6) ◽  
pp. 1163-1175 ◽  
Author(s):  
Jingyi Chen ◽  
Elton P. Hsu
Keyword(s):  
Ricci Curvature ◽  
Harmonic Functions ◽  
Volume Growth ◽  
Gradient Estimate ◽  
Gradient Estimates ◽  
Smooth Manifolds ◽  
Lipschitz Continuous ◽  
Bounded Below

AbstractWe introduce a distributional Ricci curvature on complete smooth manifolds with Lipschitz continuous metrics. Under an assumption on the volume growth of geodesics balls, we obtain a gradient estimate for weakly harmonic functions if the distributional Ricci curvature is bounded below.

Sign in / Sign up

Export Citation Format

Share Document