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Local geometric properties of conductive transmission eigenfunctions and applications

Huaian Diao, Xiaoxu Fei, Hongyu Liu

TL;DR

The paper addresses the problem of understanding how conductive transmission eigenfunctions behave near geometric corners and leverages these spectral properties to solve a geometrical inverse scattering task from minimal data. It employs Complex Geometric Optics (CGO) solutions for the perturbed operator $\Delta+(1+V)$ and Herglotz wave approximations to obtain sharp vanishing results at planar, conic, and cuboid corners in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$, even under weakened regularity assumptions. These vanishing results underpin new uniqueness statements for recovering the scatterer’s shape and its boundary impedance $\eta$ from a single far-field measurement, including local and global results under convexity-type conditions and for corona-shaped domains. The work thus connects geometric spectral properties with inverse scattering, enabling reconstruction with highly limited data in rough media and informing practical imaging of conductive layers. The methods blend CGO analysis with Herglotz extension and yield robust visibility results that rule out non-scattering when corners are present, thereby establishing sharp identifiability in a broad geometric setting.

Abstract

The purpose of the paper is twofold. First, we show that partial-data transmission eigenfunctions associated with a conductive boundary condition vanish locally around a polyhedral or conic corner in $\mathbb{R}^n$, $n=2,3$. Second, we apply the spectral property to the geometrical inverse scattering problem of determining the shape as well as its boundary impedance parameter of a conductive scatterer, independent of its medium content, by a single far-field measurement. We establish several new unique recovery results. The results extend the relevant ones in [30] in two directions: first, we consider a more general geometric setup where both polyhedral and conic corners are investigated, whereas in [30] only polyhedral corners are concerned; second, we significantly relax the regularity assumptions in [30] which is particularly useful for the geometrical inverse problem mentioned above. We develop novel technical strategies to achieve these new results.

Local geometric properties of conductive transmission eigenfunctions and applications

TL;DR

The paper addresses the problem of understanding how conductive transmission eigenfunctions behave near geometric corners and leverages these spectral properties to solve a geometrical inverse scattering task from minimal data. It employs Complex Geometric Optics (CGO) solutions for the perturbed operator and Herglotz wave approximations to obtain sharp vanishing results at planar, conic, and cuboid corners in and , even under weakened regularity assumptions. These vanishing results underpin new uniqueness statements for recovering the scatterer’s shape and its boundary impedance from a single far-field measurement, including local and global results under convexity-type conditions and for corona-shaped domains. The work thus connects geometric spectral properties with inverse scattering, enabling reconstruction with highly limited data in rough media and informing practical imaging of conductive layers. The methods blend CGO analysis with Herglotz extension and yield robust visibility results that rule out non-scattering when corners are present, thereby establishing sharp identifiability in a broad geometric setting.

Abstract

The purpose of the paper is twofold. First, we show that partial-data transmission eigenfunctions associated with a conductive boundary condition vanish locally around a polyhedral or conic corner in , . Second, we apply the spectral property to the geometrical inverse scattering problem of determining the shape as well as its boundary impedance parameter of a conductive scatterer, independent of its medium content, by a single far-field measurement. We establish several new unique recovery results. The results extend the relevant ones in [30] in two directions: first, we consider a more general geometric setup where both polyhedral and conic corners are investigated, whereas in [30] only polyhedral corners are concerned; second, we significantly relax the regularity assumptions in [30] which is particularly useful for the geometrical inverse problem mentioned above. We develop novel technical strategies to achieve these new results.
Paper Structure (12 sections, 32 theorems, 181 equations)

This paper contains 12 sections, 32 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Mathematical setup and summary of major findings
  3. Connection to existing studies and discussions
  4. Preliminaries
  5. Vanishing of transmission eigenfunctions near a convex planar corner
  6. Proof of Theorem \ref{['thm:2D']}
  7. Proof of Corollary \ref{['cor1']}
  8. Vanishing of transmission eigenfunctions near a convex conic corner or polyhedral corner
  9. Proof of Theorem \ref{['3thm:v0']}
  10. Proof of Theorem \ref{['3:cubiod']}
  11. Proof of Corollary \ref{['cor2']}
  12. Visibility and unique recovery results for the inverse scattering problem

Key Result

Proposition 2.1

Suppose that $q \in H^{1,1+\epsilon_{0}}$, where $0<\epsilon_{0}<1$. It holds that where $C$ is a positive constant and $1<\tilde{p}<2$ satisfies

Theorems & Definitions (54)

  • Proposition 2.1
  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Corollary 3.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 44 more