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 $L^2(\Omega ; d^nx)$-realizations, $n\in \mathbb {N}$, $n\geq 2$, of divergence form elliptic partial differential expressions $L$ with (nonlocal) Robin-type boundary conditions in bounded Lipschitz domains $\Omega \subset \mathbb {R}^n$, 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 \[ \nu \cdot A\nabla u + \Theta \big [u\big |_{\partial \Omega }\big ]=0 \text { on } \partial \Omega , \] where $\Theta$ represents an appropriate operator acting on Sobolev spaces associated with the boundary $\partial \Omega$ of $\Omega$, and $\nu$ denotes the outward pointing normal unit vector on $\partial \Omega$.