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A ill-posed scattering problem saturating Weyl's law

T. Chaumont-Frelet

TL;DR

This work investigates ill-posedness in three-dimensional time-harmonic scattering for the Helmholtz equation with rough coefficients. Building on Filinov's result, it constructs a compactly supported, uniformly elliptic coefficient by packing scaled copies of a reference coefficient into disjoint cubes, yielding a countable set of resonant frequencies accumulating at infinity. The main contribution shows that for any ε > 0, the number of ill-posed frequencies below ω grows like ω^{3−ε}, effectively saturating Weyl's law up to ε. The construction relies on a cube-packing lemma and a scalable, translational design that preserves ellipticity and compact support while destroying unique continuation at selected frequencies.

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 $ω> 0$ scales as $ω^3$. 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 $ω$. 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 $ω> 0$ scales as $ω^{3-\varepsilon}$.

A ill-posed scattering problem saturating Weyl's law

TL;DR

This work investigates ill-posedness in three-dimensional time-harmonic scattering for the Helmholtz equation with rough coefficients. Building on Filinov's result, it constructs a compactly supported, uniformly elliptic coefficient by packing scaled copies of a reference coefficient into disjoint cubes, yielding a countable set of resonant frequencies accumulating at infinity. The main contribution shows that for any ε > 0, the number of ill-posed frequencies below ω grows like ω^{3−ε}, effectively saturating Weyl's law up to ε. The construction relies on a cube-packing lemma and a scalable, translational design that preserves ellipticity and compact support while destroying unique continuation at selected frequencies.

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 scales as . On the other hand, scattering problems set in 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 . In this contribution, we use this result to show that for all 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 scales as .
Paper Structure (13 sections, 14 theorems, 83 equations)

This paper contains 13 sections, 14 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminary results
  3. Scalar, vector and tensors
  4. Function spaces
  5. Cubes and affine maps
  6. Reference coefficient
  7. Cube packing
  8. Main result
  9. Cube packing
  10. Construction of the corners
  11. Plan of the proof
  12. Preliminary results
  13. Proof of Lemma \ref{['lemma_cube_packing']}

Key Result

Proposition 2.1

There exists a uniformly bounded and elliptic, continuous symmetric tensor-valued coefficient $\widehat{\underline{\boldsymbol A}} \in \underline{\boldsymbol L}^\infty(\mathbb R^3)$ with $\operatorname{supp}(\widehat{\underline{\boldsymbol A}}-\underline{\boldsymbol I}) \subset \widehat{Q}$ and a fu for some $\widehat{\lambda} > 0$. In addition, we have $\widehat{\underline{\boldsymbol A}} \in \un

Theorems & Definitions (26)

  • Proposition 2.1: Reference coefficient
  • Lemma 2.2: Cube packing
  • Remark 2.3: Dyadic partition
  • Lemma 3.1: Coefficient
  • proof
  • Theorem 3.2: Compactly supported eigenfunctions
  • proof
  • Corollary 3.3: Ill-posed scattering problem
  • Proposition 4.1: Sum bound
  • proof
  • ...and 16 more