Optimal trace norms for Helmholtz problems

TL;DR

This paper rigorously analyzes weighted trace norms for Helmholtz problems, revealing their dependence on geometry and wavenumber, and providing improved boundary operator estimates that remain stable as the wavenumber approaches zero.

Contribution

It offers an explicit characterization of weighted trace norms in Helmholtz problems, including conditions for their intrinsic nature and stability estimates independent of the wavenumber.

Findings

Weighted trace norms are characterized by weighted Sobolev-Slobodeckij norms.

Trace inequalities are established with explicit wavenumber dependence.

Boundary integral operators exhibit improved continuity estimates unaffected by the wavenumber approaching zero.

Abstract

The natural $H^1(\Omega)$ energy norm for Helmholtz problems is weighted with the wavenumber modulus $\sigma$ and induces natural weighted norms on the trace spaces $H^{\pm1/2}(\Gamma)$ by minimial extension to $\Omega\subset\mathbb R^n$. This paper presents a rigorous analysis for these trace norms with an explicit characterisation by weighted Sobolev-Slobodeckij norms and scaling estimates, highlighting their dependence on the geometry of the extension set $\Omega\subset\mathbb R^n$ and the weight $\sigma$. The analysis identifies conditions under which these trace norms are intrinsic to the isolated boundary component $\Gamma\subset\partial\Omega$ and provides $\sigma$-explicit estimates for trace inequalities in weighted spaces. In these natural wavenumber-weighted norms, the boundary integral operators allow improved continuity estimates that do \emph{not} deterioriate as $\sigma\to 0$.