We present some recent results on kernel bounds for the semigroup
generated by the Dirichlet-to-Neumann operator when the
underlying operator has H\"older continuous coefficients and the
domain has a $C^{1+\kappa}$-boundary. The proof
depends on Gaussian bounds for derivatives of the semigroup
kernel of an elliptic operator with Dirichlet boundary conditions.
As a consequence the Dirichlet-to-Neumann semigroup is holomorphic
on the right half-plane on $L_1$.
If time permits, we also discuss the diamagnetic Dirichlet-to-Neumann
operator.
This is joint work with El Maati Ouhabaz.
References.
\begin{itemize}
\item
{\sc Elst, A. F.~M. ter {\rm and} Ouhabaz, E.M.}, Dirichlet-to-Neumann and
elliptic operators on $C^{1+\kappa}$-domains: Poisson and Gaussian bounds.
{\em J. Differential Equations} {\bf 267} (2019), 4224--4273.
\item
{\sc Elst, A. F.~M. ter {\rm and} Ouhabaz, E.M.}, Analyticity of the
Dirichlet-to-Neumann semigroup on continuous functions.
{\em J. Evol. Eq.} {\bf 19} (2019), 21--31.
\item
{\sc Elst, A. F.~M. ter {\rm and} Ouhabaz, E.M.}, The diamagnetic inequality
for the Dirichlet-to-Neumann operator.
{\em Bull. Lond. Math. Soc.} (2022).
DOI: 10.1112/blms.12674.
\end{itemize}