Analyticity of the Dirichlet-to-Neumann semigroup on continuous functions

Abstract

Let $\Omega$ be a bounded open subset with $C^{1+\kappa }$-boundary for some $\kappa > 0$. Consider the Dirichlet-to-Neumann operator associated with the elliptic operator $- \sum \partial _l ( c_{kl} \, \partial _k ) + V$, where the $c_{kl} = c_{lk}$ are Hölder continuous and $V \in L_\infty (\Omega )$ are real valued. We prove that the Dirichlet-to-Neumann operator generates a $C_0$-semigroup on the space $C(\partial \Omega )$ which is in addition holomorphic with angle $\frac{\pi }{2}$. We also show that the kernel of the semigroup has Poisson bounds on the complex right half-plane. As a consequence, we obtain an optimal holomorphic functional calculus and maximal regularity on $L_p(\Gamma )$ for all $p \in (1,\infty )$.