The Dirichlet-to-Neumann operator on rough domains

TL;DR

This paper investigates the Dirichlet-to-Neumann operator on rough domains with finite boundary measure, analyzing its spectral properties and long-term behavior of associated semigroups in relation to boundary trace characteristics.

Contribution

It defines the Dirichlet-to-Neumann operator on irregular domains via form methods and studies its self-adjointness, semigroup generation, and asymptotic behavior.

Findings

The operator $-D_0$ is self-adjoint and generates a contractive $C_0$-semigroup.

The asymptotic behavior of the semigroup relates to boundary trace properties of functions.

The analysis links geometric boundary features to spectral and dynamical properties.

Abstract

We consider a bounded connected open set $\Omega \subset {\rm R}^d$ whose boundary $\Gamma$ has a finite $(d-1)$-dimensional Hausdorff measure. Then we define the Dirichlet-to-Neumann operator $D_0$ on $L_2(\Gamma)$ by form methods. The operator $-D_0$ is self-adjoint and generates a contractive $C_0$-semigroup $S = (S_t)_{t > 0}$ on $L_2(\Gamma)$. We show that the asymptotic behaviour of $S_t$ as $t \to \infty$ is related to properties of the trace of functions in $H^1(\Omega)$ which $\Omega$ may or may not have.