Integral equation methods for scattering by general compact obstacles: wavenumber-explicit estimates
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arXiv
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| Format: | Preprint |
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2026
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| author | Chandler-Wilde, Simon N. Sadeghi, Siavash |
| author_facet | Chandler-Wilde, Simon N. Sadeghi, Siavash |
| contents | 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 $d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_19456 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Integral equation methods for scattering by general compact obstacles: wavenumber-explicit estimates Chandler-Wilde, Simon N. Sadeghi, Siavash Analysis of PDEs 45B05, 31B10, 35P25, 78A45, 28A80 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 $d$. |
| title | Integral equation methods for scattering by general compact obstacles: wavenumber-explicit estimates |
| topic | Analysis of PDEs 45B05, 31B10, 35P25, 78A45, 28A80 |
| url | https://arxiv.org/abs/2601.19456 |