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

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