Estimates of heat kernels with Neumann boundary conditions
type: Item
instance of: scholarly article
title: Estimates of heat kernels with Neumann boundary conditions
zbMATH Open document ID: 1262.35115
author: Xue Yang
author: Tu-Sheng Zhang
published in: Potential Analysis
publication date: 2013-02-15T00:00:00Z
review text: This paper is concerned with two-sided estimates for the heat kernels corresponding to general elliptic operators of the form \(Lu=\frac{1}{2}\nabla\cdot(A\nabla u)+b\cdot \nabla u-\nabla(\hat bu)+qu\) in a bounded domain \(D\subset \mathbb R^N\) subject to Robin boundary conditions: \(\frac{1}{2}\langle A\nabla u,n\rangle-\langle \hat b,n\rangle u=0\) on \(\partial D\). Here \(A\) is a smooth, symmetric and uniformly elliptic matrix-valued function, \(b\), \(\hat b\) are Borel measurable vector valued function. Using a time-reverse technique the authors translate the problem into estimating the fundamental solution of the simpler problem \(u_t=\frac{1}{2}\nabla\cdot(A\nabla u)+\tilde b\cdot \nabla u+\tilde qu\) in \(D\) subject to the Neumann boundary condition \(\frac{1}{2}\langle A\nabla u,n\rangle=0\) on \(\partial D\). The upper bound on the fundamental solution is obtained by using a parametrix and perturbation methods, whereas the lower bound is obtained under the assumption of convexity on \(D\).
reviewed by: Marius Ghergu
Mathematics Subject Classification ID: 35K08
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 6136723
zbMATH Keywords: Dirichlet forms
zbMATH Keywords: time-reverse technique
MaRDI profile type: Publication
full work available at URL: s11118-012-9286-9
OpenAlex ID: W2158780929
cites work: Bounds for the fundamental solution of a parabolic equation
cites work: Local behavior of solutions of quasilinear parabolic equations
cites work: Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains
cites work: Time-reversal and elliptic boundary value problems
cites work: Perturbation of symmetric Markov processes
cites work: A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash
cites work: Dirichlet forms and symmetric Markov processes
cites work: A boundary value problem of partial differential equations of parabolic type
cites work: Integrability of functionals of Dirichlet processes, probabilistic representations of semigroups, and estimates of heat kernels
cites work: Multi-dimensional diffusion and the Markov process on the boundary
cites work: A Harnack inequality for the equation \(\nabla (a \nabla u) + b \nabla u = 0\), when \(| b | \in K_{n+1}\)
cites work: Gaussian bounds for the fundamental solutions of \(\nabla(A\nabla u)+B\nabla u-u_ t=0\)
DOI: 10.1007/S11118-012-9286-9
description: scientific article; zbMATH DE number 6136723