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

TL;DR

This paper proves that the Dirichlet-to-Neumann operator generates a holomorphic semigroup on continuous boundary functions, with implications for functional calculus and regularity in elliptic PDEs.

Contribution

It establishes the analyticity of the Dirichlet-to-Neumann semigroup on continuous functions and derives optimal bounds and regularity results.

Findings

Semigroup is holomorphic with angle π/2

Kernel has Poisson bounds on complex right half-plane

Achieves optimal functional calculus and maximal regularity

Abstract

Let $\Omega$ be a bounded open subset with $C^{1+\kappa}$-boundary for some $\kappa > 0$. Consider the Dirichlet-to-Neumann operator associated to the elliptic operator $- \sum \partial_l ( c_{kl} \, \partial_k ) + V$, where the $c_{kl} = c_{lk}$ are H\"older 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 $p \in (1,\infty)$.