Optimal trace norms for Helmholtz problems

Abstract

The natural $H^1(\Omega)$ energy norm for Helmholtz problems is weighted with the wavenumber modulus $\sigma$ and induces weighted norms on the trace spaces $H^{\pm1/2}(\Gamma)$ by minimal extension to $\Omega\subset\mathbb R^n$. This paper provides an explicit characterisation through weighted Sobolev-Slobodeckij norms and scaling estimates, highlighting the 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 establishes $\sigma$-explicit trace estimates in weighted spaces. In these norms, the Helmholtz potential and boundary integral operators satisfy improved coercivity and continuity estimates without additional low-frequency factors that deteriorate as $\sigma\to 0$; for $n\geq 3$, the same analysis also improves the corresponding bounds in the classical unweighted trace norms.