Heat Kernel Bounds for Elliptic Partial Differential Operators in Divergence Form with Robin-Type Boundary Conditions

One of the principal topics of this paper concerns the realization of self-adjoint operators $L_{\mathrm{\Theta },\text{Om}}$ in $L^2(\text{Om};d^{n}x{)}^m$, $m, n \in \bbN$, associated with divergence form elliptic partial differential expressions $L$ with (nonlocal) Robin-type boundary conditions in bounded Lipschitz domains $\text{Om}\subset {\text{bbR}}^n$. In particular, we develop the theory in the vector-valued case and hence focus on matrix-valued differential expressions $L$ which act as $$Lu=-\left(\sum _{j,k=1}^n{\mathrm{\partial }}_j\right(\sum _{\beta =1}^m{a}_{j,k}^{\alpha ,\beta }{\mathrm{\partial }}_k{u}_{\beta }){)}_{1\le \alpha \le m},u=(u_1,\mathrm{.}\mathrm{.}\mathrm{.},u_m)\mathrm{.}$$ The (nonlocal) Robin-type boundary conditions are then of the form $$\nu \cdot ADu+\mathrm{\Theta }\left[u{\mid }_{\mathrm{\partial }\text{Om}}\right]=0\text{on }\mathrm{\partial }\text{Om},$$ where $\mathrm{\Theta }$ represents an appropriate operator acting on Sobolev spaces associated with the boundary $\mathrm{\partial }\text{Om}$ of $\text{Om}$, $\nu$ denotes the outward pointing normal unit vector on $\mathrm{\partial }\text{Om}$, and $Du:=\bigl(\partial_j u_\alpha\bigr){\substack{1\leq\alpha\leq m 1\leq j\leq n}}$. Assuming $\mathrm{\Theta }\ge 0$ in the scalar case $m=1$, we prove Gaussian heat kernel bounds for $L_{\mathrm{\Theta },\text{Om}}$ by employing positivity preserving arguments for the associated semigroups and reducing the problem to the corresponding Gaussian heat kernel bounds for the case of Neumann boundary conditions on $\mathrm{\partial }\text{Om}$. We also discuss additional zero-order potential coefficients $V$ and hence operators corresponding to the form sum $L{\Theta, \Om} + V$.