Integral equation methods for scattering by general compact obstacles: wavenumber-explicit estimates

TL;DR

The article develops wavenumber-explicit bounds for Caetano et al.'s boundary integral equation $A_k$ governing exterior Helmholtz scattering by general compact obstacles, extending beyond Lipschitz geometries to $d$-set fractal boundaries. It proves $k$-dependent bounds $\|A_{k,k}\|_k \lesssim k$ and, away from spectral obstructions, $\|A_{k,k}^{-1}\|_k \lesssim k^2\big(C_{k,R}(\Omega)+C_k(\Omega_- )\big)$, with sharper or alternative growth depending on geometry (e.g., star-shaped obstacles) and exceptional sets of small measure where polynomial growth is still controlled. The work highlights that the inverse norms can grow arbitrarily fast along carefully constructed sequences of obstacles but remain polynomial outside a small exceptional set, tying conditioning to geometry and spectral data. Consequently, it yields first $k$-explicit bounds for the classical single-layer operator $S_k$ and its inverse in Lipschitz-domain and screen settings, and extends the analysis to $d$-sets, broadening the applicability of boundary-integral methods to complex and fractal geometries with precise wavenumber behavior.

Abstract

There has been significant recent interest in understanding the dependence on the wavenumber, $k$, of boundary integral operators (BIOs), supported on some set $Γ\subset \mathbb{R}^n$, that arise in the solution of BVPs for the Helmholtz equation, $Δu + k^2 u=0$. Recently, for the Dirichlet BVP with data $g$, Caetano et al (2025) have proposed an integral equation (IE) $A_kφ=g$ that applies for arbitrary compact $Γ$. This formulation is a generalisation of standard first kind IEs, where the BIO is $S_k$, the single-layer BIO on a surface $Γ$, that apply when $Γ$ is the boundary of a Lipschitz domain or a screen. In this paper we study the dependence of $A_k$ on $k$, showing that, for $k\geq k_0>0$, $\|A_k\|\leq ck$ while $\|A_k^{-1}\| \leq c'k$ if $Γ$ is star-shaped, where $c, c'>0$ depend only on $k_0$ and $Γ$. Amongst other bounds we also show that: (i) on the one hand, given any mildly increasing unbounded positive sequence $(k_m)$ and any unbounded sequence $(a_m)$, there exists $Γ$, with connected complement, such that $\|A_{k_m}^{-1}\|\geq a_m$ for every $m$; (ii) on the other hand, for every $Γ\subset \mathbb{R}^n$ and $k_0,\varepsilon, δ>0$, there exists $c>0$ and $E\subset [k_0,\infty)$, with Lebesgue measure $m(E)\leq \varepsilon$, such that $\|A_{k}^{-1}\|\leq c k^{2n+2+δ}$ on $[k_0,\infty)\setminus E$, i.e., the growth of $\|A_{k}^{-1}\|$ is at worst polynomial in $k$ if one avoids a set $E$ of arbitrarily small measure. As a corollary of these results we obtain the first $k$-explicit bounds on $\|S_k^{-1}\|$ and the condition number of $S_k$ for the case that $Γ$ is the boundary of a Lipschitz domain, or a screen not contained in a hyperplane, and analogous estimates for the case that $Γ$ is a $d$-set (and so of Hausdorff dimension $d$), for non-integer values of $d$.

Figures (1)

• Figure 1: Geometry schematic. The compact obstacle $O$ is shaded in gray, and $\Omega= O^c:= \mathbb{R}^n\setminus O$ is its connected complement. $\Gamma$ (thick blue lines) is (one possible choice of) the compact set, satisfying \ref{['eq:GamRest']}, on which the IE \ref{['eq:iemain']} is posed, and $\Omega_-:=O\setminus \Gamma$, so that $O=\Gamma\cup \Omega_-$ and $\Omega\cup \Omega_-=\Gamma^c$.

Theorems & Definitions (43)

• Proposition 2.1
• Proposition 2.2
• Theorem 2.3
• Definition 2.4: Star-shaped
• Corollary 2.5
• Corollary 2.6
• Proposition 2.7
• Corollary 2.8
• Remark 2.9: 1st vs. 2nd kind IEs
• Remark 2.10: Previous work on 1st kind IEs