Hölder 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 $Ω\subset {\bf R}^d$ with Lipschitz boundary conditions, where $c_{kl} \in L_\infty(Ω,{\bf R})$ and $a_k,b_k,a_0 \in L_\infty(Ω,{\bf C})$, subject to Robin boundary conditions $\partial_νu + β\, {\rm Tr}\, u = 0$, where $β\in L_\infty(\partial Ω, {\bf C})$ is complex valued. Then we show that the kernel of the semigroup generated by $-A$ satisfies Gaussian estimates and Hölder Gaussian estimates. If all coefficients and the function $β$ are real valued, then we prove Gaussian lower bounds. Finally, if $Ω$ is of class $C^{1+κ}$ with $κ> 0$, $c_{kl} = c_{lk}$ is Hölder 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 Hölder Poisson bounds.