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 $\Omega \subset {\mathbf{R}}^d$ with Lipschitz boundary conditions, where $c_{kl}\in L_{\infty }(\Omega ,\mathbf{R})$ and $a_k,b_k,a_0\in L_{\infty }(\Omega ,\mathbf{C})$, subject to Robin boundary conditions ${\partial }_{\nu }u+\beta \mathrm{Tr}u=0$, where $\beta \in L_{\infty }(\partial \Omega ,\mathbf{C})$ 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 $\beta$ 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 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.

References in corpus (2)

• Regularity of Solutions of Linear Second Order Elliptic and Parabolic Boundary Value Problems on Lipschitz Domains
• The Dirichlet-to-Neumann operator via hidden compactness