The Dirichlet-to-Neumann operator on $C(\partial \Omega)$

TL;DR

This paper studies the Dirichlet-to-Neumann operator on Lipschitz boundaries, showing the associated semigroup preserves continuity, exploring positivity, spectral properties, and conditions for having a continuous kernel.

Contribution

It demonstrates that the semigroup generated by the Dirichlet-to-Neumann operator on $C(oundary \, ext{domain})$ is strongly continuous and investigates its positivity and spectral characteristics.

Findings

The semigroup leaves $C(oundary \, ext{domain})$ invariant.

The semigroup is strongly continuous on $C(oundary \, ext{domain})$.

New criteria for semigroups to possess a continuous kernel.

Abstract

Let $\Omega \subset {\bf R}^d$ be an open bounded set with Lipschitz boundary $\Gamma$. Let $D_V$ be the Dirichlet-to-Neumann operator with respect to a purely second-order symmetric divergence form operator with real Lipschitz continuous coefficients and a positive potential $V$. We show that the semigroup generated by $-D_V$ leaves $C(\Gamma)$ invariant and that the restriction of this semigroup to $C(\Gamma)$ is a $C_0$-semigroup. We investigate positivity and spectral properties of this semigroup. We also present results where $V$ is allowed to be negative. Of independent interest is a new criterium for semigroups to have a continuous kernel.