Optimal trace norms for Helmholtz problems

Abstract

The natural $H^1(\hat{I}\text{c◯})$ energy norm for Helmholtz problems is weighted with the wavenumber modulus $\ddot{I}ƒ$ and induces weighted norms on the trace spaces $H^{\pm 1/2}(\hat{I}“)$ by minimal extension to $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$. This paper provides an explicit characterisation through weighted Sobolev-Slobodeckij norms and scaling estimates, highlighting the dependence on the geometry of the extension set $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$ and the weight $\ddot{I}ƒ$. The analysis identifies conditions under which these trace norms are intrinsic to the isolated boundary component $\hat{I}“\subset \partial \hat{I}\text{c◯}$ and establishes $\ddot{I}ƒ$-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 $\ddot{I}ƒ\to 0$; for $n\ge 3$, the same analysis also improves the corresponding bounds in the classical unweighted trace norms.