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

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, $n\in \mathbb{N}$, $n\ge 2$, 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 $$ ν\cdot A\nabla u + Θ\big[u\big|_{\partial\Om}\big]=0 \, \text{on} \, \partialΩ, $$ 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 $\partial\Om$.

arXiv comment14 pages