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

There has been significant recent interest in understanding the dependence on the wavenumber, $k$, of boundary integral operators (BIOs), supported on some set $\Gamma\subset \mathbb{R}^n$, that arise in the solution of the Helmholtz equation, $\Delta u + k^2 u=0$.

Recently, for the Dirichlet boundary value problem with data $g$, Caetano et al (Proc.

R.

Soc.

A, 481:20230650, 2025) have proposed a novel integral equation $A_k\phi=g$ that applies for arbitrary compact $\Gamma$.

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 $\Gamma$ is star-shaped, where $c, c'>0$ depend only on $k_0$ and $\Gamma$.

Amongst other bounds we show that: (i) on the one hand, given any mildly increasing unbounded positive sequence $(k_m)$ and any unbounded sequence $(a_m)$, there exists $\Gamma$, with connected complement, such that $\|A_{k_m}^{-1}\|\geq a_m$ for every $m$; (ii) on the other hand, for every $\Gamma\subset \mathbb{R}^n$ and $k_0,\varepsilon, \delta>0$, there exists $c>0$ and $E\subset [k_0,\infty)$, with Lebesgue measure $m(E)\leq \varepsilon$, such that $\|A_{k}^{-1}\|\leq c k^{2n+2+\delta}$ 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 we obtain the first $k$-explicit bounds on the condition number of $S_k$, where $S_k$ is the standard single-layer BIO on $\Gamma$ when $\Gamma$ is the boundary of a Lipschitz domain, and analogous estimates when $\Gamma$ is a $d$-set (and so of Hausdorff dimension $d$), for non-integer values of $d$.