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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 $L_{Θ, \Om}$ in $L^2(\Om; d^n x)^m$, $m, n \in \bbN$, associated with divergence form elliptic partial differential expressions $L$ with (nonlocal) Robin-type boundary conditions in bounded Lipschitz domains $\Om \subset \bbR^n$. In particular, we develop the theory in the vector-valued case and hence focus on matrix-valued differential expressions $L$ which act as $$ Lu = - \biggl(\sum_{j,k=1}^n\partial_j\bigg(\sum_{β= 1}^m a^{α,β}_{j,k}\partial_k u_β\bigg) \bigg)_{1\leqα\leq m}, \quad u=(u_1,...,u_m). $$ The (nonlocal) Robin-type boundary conditions are then of the form $$ ν\cdot A D u + Θ\big[u\big|_{\partial \Om}\big] = 0 \, \text{on $\partial \Om$}, $$ where $Θ$ represents an appropriate operator acting on Sobolev spaces associated with the boundary $\partial \Om$ of $\Om$, $ν$ denotes the outward pointing normal unit vector on $\partial\Om$, and $Du:=\bigl(\partial_j u_α\bigr)_{\substack{1\leqα\leq m 1\leq j\leq n}}$. Assuming $Θ\geq 0$ in the scalar case $m=1$, we prove Gaussian heat kernel bounds for $L_{Θ, \Om}$ 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 $\partial \Om$. We also discuss additional zero-order potential coefficients $V$ and hence operators corresponding to the form sum $L_{Θ, \Om} + V$.