H\"older kernel estimates for Robin operators and Dirichlet-to-Neumann operators

Abstract

Consider the elliptic operator \[ A = - \sum_{k,l=1}^d \partial_k \, c_{kl} \, \partial_l + \sum_{k=1}^d a_k \, \partial_k - \sum_{k=1}^d \partial_k \, b_k + a_0 \] on a bounded connected open set $\Omega \subset {\bf R}^d$ with Lipschitz boundary conditions, where $c_{kl} \in L_\infty(\Omega,{\bf R})$ and $a_k,b_k,a_0 \in L_\infty(\Omega,{\bf C})$, subject to Robin boundary conditions $\partial_\nu u + \beta \, {\rm Tr}\, u = 0$, where $\beta \in L_\infty(\partial \Omega, {\bf C})$ is complex valued. Then we show that the kernel of the semigroup generated by $-A$ satisfies Gaussian estimates and H\"older Gaussian estimates. If all coefficients and the function $\beta$ are real valued, then we prove Gaussian lower bounds. Finally, if $\Omega$ is of class $C^{1+\kappa}$ with $\kappa > 0$, $c_{kl} = c_{lk}$ is H\"older continuous, $a_k = b_k = 0$ and $a_0$ is real valued, then we show that the kernel of the semigroup associated to the Dirichlet-to-Neumann operator corresponding to $A$ has H\"older Poisson bounds.