A ill-posed scattering problem saturating Weyl's law

TL;DR

This paper constructs examples of scattering problems with rough coefficients that are ill-posed at infinitely many frequencies, with the number of problematic frequencies growing nearly as fast as Weyl's law predicts for smooth domains.

Contribution

It demonstrates that scattering problems with rough coefficients can be ill-posed at infinitely many frequencies, nearly matching Weyl's law growth rate, extending Filinov's counterexample.

Findings

Ill-posed scattering at infinitely many frequencies.

Number of problematic frequencies scales as ω^{3-ε}.

Constructs rough coefficients causing ill-posedness.

Abstract

This paper focuses on the well-posedness (or lack thereof) of three-dimensional time-harmonic wave propagation problems modeled by the Helmholtz equation. It is well-known that if the problem is set in bounded domain with Dirichlet boundary conditions, then the Helmholtz problem is well-posed for all (real-valued) frequencies except for a sequence of countably many resonant frequencies that accumulate at infinity. In fact, if the domain is sufficiently smooth, this can be quantified further and Weyl's law states that the number of resonant frequencies less than a given $\omega > 0$ scales as $\omega^3$. On the other hand, scattering problems set in $\mathbb{R}^3$ with a radiation condition at infinity and a bounded obstacle modeled by variations in the PDE coefficients are well-posed for all frequencies under mild regularity assumption on such coefficients. In 2001, Filinov provided a counter example of a rough coefficient such that the scattering problem is not well-posed for (at least) a single frequency $\omega$. In this contribution, we use this result to show that for all $\varepsilon > 0$ we can design a rough coefficient corresponding to a compactly supported obstacle such that the scattering problem is ill-posed for a countable sequence of frequencies accumulating at infinity, and such that the number of such frequencies less than any given $\omega > 0$ scales as $\omega^{3-\varepsilon}$.