Heat kernel bounds for elliptic partial differential operators in divergence form with Robin-type boundary conditions II

TL;DR

This paper extends Gaussian heat kernel bounds for divergence form elliptic operators with Robin-type boundary conditions in Lipschitz domains, broadening understanding of heat distribution in complex boundary settings.

Contribution

It generalizes recent results by including nonlocal Robin boundary conditions for elliptic operators in Lipschitz domains.

Findings

Established Gaussian heat kernel bounds for operators with Robin-type boundary conditions.

Extended existing bounds to nonlocal boundary conditions.

Applicable to bounded Lipschitz domains in bR^n.

Abstract

The principal aim of this short note is to extend a recent result on Gaussian heat kernel bounds for self-adjoint $L^2(\Om; d^n x)$-realizations, $n\in\bbN$, $n\geq 2$, of divergence form elliptic partial differential expressions $L$ with (nonlocal) Robin-type boundary conditions in bounded Lipschitz domains $\Om \subset \bbR^n$, where $$ Lu = - \sum_{j,k=1}^n\partial_j a_{j,k}\partial_k u. $$ The (nonlocal) Robin-type boundary conditions are then of the form $$ \nu\cdot A\nabla u + \Theta\big[u\big|_{\partial\Om}\big]=0 \, \text{on} \, \partial\Omega, $$ where $\Theta$ represents an appropriate operator acting on Sobolev spaces associated with the boundary $\partial \Om$ of $\Om$, and $\nu$ denotes the outward pointing normal unit vector on $\partial\Om$.