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

TL;DR

This paper provides explicit bounds on the dependence of boundary integral operators on the wavenumber in Helmholtz problems, revealing polynomial growth and constructing obstacles with specific inverse operator behaviors.

Contribution

It introduces wavenumber-explicit estimates for integral operators on general obstacles, including bounds on their norms and inverses, extending previous results to more general geometries.

Findings

Bounded the operator norm as proportional to k for large k

Constructed obstacles with inverse norms growing arbitrarily fast

Showed inverse norms grow at most polynomially outside small measure sets

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 $d$.