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On a novel UCP result and its application to inverse conductive scattering

Huaian Diao, Xiaoxu Fei, Hongyu Liu

TL;DR

This work addresses the inverse conductive medium scattering problem by establishing a novel UCP for a two-component elliptic PDE system with transmission across an irrational corner, then leveraging CGO methods and corner microlocal analysis to derive strong identifiability results. The authors prove that convex irrational corners cannot be invisible and that a single far-field measurement suffices to uniquely identify convex irrational polygonal scatterers, as well as polygonal-nest and polygonal-cell structures along with their material parameters, under suitable admissible conditions. The approach fuses Complex Geometric Optics (CGO) with Fourier-type expansions near corners to extract sharp information from limited data, advancing Schiffer-type identifiability in conductive media and offering techniques potentially applicable to broader elliptic systems. Collectively, the results provide new theoretical guarantees for shape and parameter recovery from minimal measurements in two-dimensional conductive scattering problems, and they introduce analytical tools with possible broader impact in inverse problems and PDE analysis.

Abstract

In this paper, we derive a novel Unique Continuation Principle (UCP) for a system of second-order elliptic PDEs system and apply it to investigate inverse problems in conductive scattering. The UCP relaxes the typical assumptions imposed on the domain or boundary with certain interior transmission conditions. This is motivated by the study of the associated inverse scattering problem and enables us to establish several novel unique identifiability results for the determination of generalized conductive scatterers using a single far-field pattern, significantly extending the results in [15,23]. A key technical advancement in our work is the combination of Complex Geometric Optics (CGO) techniques from [15,23] with the Fourier expansion method to microlocally analyze corner singularities and their implications for inverse problems. We believe that the methods developed can have broader applications in other contexts.

On a novel UCP result and its application to inverse conductive scattering

TL;DR

This work addresses the inverse conductive medium scattering problem by establishing a novel UCP for a two-component elliptic PDE system with transmission across an irrational corner, then leveraging CGO methods and corner microlocal analysis to derive strong identifiability results. The authors prove that convex irrational corners cannot be invisible and that a single far-field measurement suffices to uniquely identify convex irrational polygonal scatterers, as well as polygonal-nest and polygonal-cell structures along with their material parameters, under suitable admissible conditions. The approach fuses Complex Geometric Optics (CGO) with Fourier-type expansions near corners to extract sharp information from limited data, advancing Schiffer-type identifiability in conductive media and offering techniques potentially applicable to broader elliptic systems. Collectively, the results provide new theoretical guarantees for shape and parameter recovery from minimal measurements in two-dimensional conductive scattering problems, and they introduce analytical tools with possible broader impact in inverse problems and PDE analysis.

Abstract

In this paper, we derive a novel Unique Continuation Principle (UCP) for a system of second-order elliptic PDEs system and apply it to investigate inverse problems in conductive scattering. The UCP relaxes the typical assumptions imposed on the domain or boundary with certain interior transmission conditions. This is motivated by the study of the associated inverse scattering problem and enables us to establish several novel unique identifiability results for the determination of generalized conductive scatterers using a single far-field pattern, significantly extending the results in [15,23]. A key technical advancement in our work is the combination of Complex Geometric Optics (CGO) techniques from [15,23] with the Fourier expansion method to microlocally analyze corner singularities and their implications for inverse problems. We believe that the methods developed can have broader applications in other contexts.
Paper Structure (7 sections, 13 theorems, 156 equations, 4 figures)

This paper contains 7 sections, 13 theorems, 156 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The main results.
  3. The conductive medium scattering and invisibility
  4. Connections to existing results and main contributions
  5. The proofs of Theorem \ref{['thm:ucp']} and Corollaries \ref{['thm:invis']}-\ref{['thm:uni pre']}
  6. Unique recovery results for the conductive polygonal-nest and polygonal-cell medium scatterers
  7. The proofs of Theorems \ref{['2thm:un-eta-cell']} and \ref{['2thm:un-eta-nest']}

Key Result

Theorem 1.1

Let $D$ and $\Omega$ be two bounded Lipschitz domains in $\mathbb R^2$ with connected complements $\mathbb R^2\setminus \overline{D}$ and $\mathbb R^2\setminus \overline{\Omega}$ respectively, where $\overline {\Omega} \subset D$. Assume $\gamma_1\in \mathbb R_{+}$ and $\gamma_2\in L^\infty(D)\cap H where Here the constant $\eta\in \mathbb C$ is defined on $\Gamma_{r_0}^\pm$ and $\partial_\nu v$

Figures (4)

  • Figure 1: The schematic illustrations of the polygonal-nest and the polygonal-cell structures.
  • Figure 2: A schematic illustration of network for polygonal-nest structure.
  • Figure 3: Schematic illustration of the polygonal-cell structure.
  • Figure 4: Schematic illustration of the polygonal-nest structure.

Theorems & Definitions (34)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Corollary 1.2
  • Definition 1.1
  • Corollary 1.3
  • Lemma 2.1
  • Proposition 2.1
  • Lemma 2.2
  • proof : The proof of Theorem \ref{['thm:ucp']}
  • ...and 24 more