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 $\hat{I}“\subset {\mathbb{R}}^n$, that arise in the solution of the Helmholtz equation, $\hat{I}”u+k^{2}u=0$. 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 $A_k\ddot{I}•=g$ that applies for arbitrary compact $\hat{I}“$. In this paper we study the dependence of $A_k$ on $k$, showing that, for $k\ge k_0>0$, $\Vert A_k\Vert \le ck$ while $\Vert A_k^{-1}\Vert \le c^{\prime }k$ if $\hat{I}“$ is star-shaped, where $c,c^{\prime }>0$ depend only on $k_0$ and $\hat{I}“$. 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 $\hat{I}“$, with connected complement, such that $\Vert A_{k_m}^{-1}\Vert \ge a_m$ for every $m$; (ii) on the other hand, for every $\hat{I}“\subset {\mathbb{R}}^n$ and $k_0,\epsilon ,\hat{I}´>0$, there exists $c>0$ and $E\subset [k_0,\infty )$, with Lebesgue measure $m(E)\le \epsilon$, such that $\Vert A_k^{-1}\Vert \le c{k}^{2n+2+\hat{I}´}$ on $[k_0,\infty )\setminus E$, i.e., the growth of $\Vert A_k^{-1}\Vert$ 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 $\hat{I}“$ when $\hat{I}“$ is the boundary of a Lipschitz domain, and analogous estimates when $\hat{I}“$ is a $d$-set (and so of Hausdorff dimension $d$), for non-integer values of $d$.

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