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

Abstract

One of the principal topics of this paper concerns the realization of self-adjoint operators $L_{Θ, \Om}$ in $L^2(\Om; d^n x)^m$, $m,n\in \mathbb{N}$, 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 $$Lu=-\left(\sum _{j,k=1}^n{\partial }_j\right(\sum _{\beta =1}^m{a}_{j,k}^{\alpha ,\beta }{\partial }_k{u}_{\beta }){)}_{1\le \alpha \le m},u=(u_1,...,u_m).$$ 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 $\Theta$ represents an appropriate operator acting on Sobolev spaces associated with the boundary $\partial \Om$ of $\Om$, $\nu$ denotes the outward pointing normal unit vector on $\partial\Om$, and $Du:=({\partial }_j{u}_{\alpha }{)}_{\begin{array}{c}1\le \alpha \le m1\le j\le n\end{array}}$. Assuming $\Theta \ge 0$ 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 $L_{Θ, \Om} + V$.

45 pages; small corrections are made in this version