Estimates of heat kernels with Neumann boundary conditions
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 \(D\).
scientific article; zbMATH DE number 6136723