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

Abstract

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 BVPs for the Helmholtz equation, $\Delta u + k^2 u=0$. Recently, for the Dirichlet BVP with data $g$, Caetano et al (2025) have proposed an integral equation (IE) $A_k\phi=g$ that applies for arbitrary compact $\Gamma$. This formulation is a generalisation of standard first kind IEs, where the BIO is $S_k$, the single-layer BIO on a surface $\Gamma$, that apply when $\Gamma$ 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 $\Gamma$ is star-shaped, where $c, c'>0$ depend only on $k_0$ and $\Gamma$. 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 $\Gamma$, 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 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 $\Gamma$ is the boundary of a Lipschitz domain, or a screen not contained in a hyperplane, and analogous estimates for the case that $\Gamma$ is a $d$-set (and so of Hausdorff dimension $d$), for non-integer values of $d$.