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The obstacle scattering for the biharmonic equation

Chengyu Wu, Jiaqing Yang

TL;DR

The paper addresses direct and inverse obstacle scattering for the biharmonic operator in ${\mathbb R}^d$ with Dirichlet boundary in $d=2,3$, introducing a new two-component far-field pattern via $u^s_\pm=\Delta u^s\pm k^2 u^s$ and a corresponding reciprocity framework. It leverages the natural decomposition $\Delta^2-k^4=(\Delta+k^2)(\Delta-k^2)$ to reduce the direct problem to Helmholtz-type equations and develops a boundary-integral equation method that yields well-posedness in both dimensions under mild boundary regularity. The inverse problem is tackled through new reciprocity relations that couple the far-field patterns and the scattered field, establishing uniqueness of obstacle recovery from fixed-frequency measurements. Overall, the work extends biharmonic scattering theory by combining boundary-integral techniques, pseudodifferential operator theory, and reciprocity to provide a rigorous, practically relevant framework for elasticity and plate vibration problems.

Abstract

In this paper, we consider the obstacle scattering problem for biharmonic equations with a Dirichlet boundary condition in both two and three dimensions. Some basic properties are first derived for the biharmonic scattering solutions, which leads to a simple criterion for the uniqueness of the direct problem. Then a new type far-field pattern is introduced, where the correspondence between the far-field pattern and scattered field is established. Based on these properties, we prove the well-posedness of the direct problem in associated function spaces by utilizing the boundary integral equation method, which relys on a natural decomposition of the biharmonic operator and the theory of the pseudodifferential operator. Furthermore, the inverse problem for determining the obstacle is studied. By establishing some novel reciprocity relations between the far-field pattern and scattered field, we show that the obstacle can be uniquely recovered from the measurements at a fixed frequency.

The obstacle scattering for the biharmonic equation

TL;DR

The paper addresses direct and inverse obstacle scattering for the biharmonic operator in with Dirichlet boundary in , introducing a new two-component far-field pattern via and a corresponding reciprocity framework. It leverages the natural decomposition to reduce the direct problem to Helmholtz-type equations and develops a boundary-integral equation method that yields well-posedness in both dimensions under mild boundary regularity. The inverse problem is tackled through new reciprocity relations that couple the far-field patterns and the scattered field, establishing uniqueness of obstacle recovery from fixed-frequency measurements. Overall, the work extends biharmonic scattering theory by combining boundary-integral techniques, pseudodifferential operator theory, and reciprocity to provide a rigorous, practically relevant framework for elasticity and plate vibration problems.

Abstract

In this paper, we consider the obstacle scattering problem for biharmonic equations with a Dirichlet boundary condition in both two and three dimensions. Some basic properties are first derived for the biharmonic scattering solutions, which leads to a simple criterion for the uniqueness of the direct problem. Then a new type far-field pattern is introduced, where the correspondence between the far-field pattern and scattered field is established. Based on these properties, we prove the well-posedness of the direct problem in associated function spaces by utilizing the boundary integral equation method, which relys on a natural decomposition of the biharmonic operator and the theory of the pseudodifferential operator. Furthermore, the inverse problem for determining the obstacle is studied. By establishing some novel reciprocity relations between the far-field pattern and scattered field, we show that the obstacle can be uniquely recovered from the measurements at a fixed frequency.
Paper Structure (5 sections, 22 theorems, 92 equations)

This paper contains 5 sections, 22 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some basic properties
  4. Well-posedness
  5. The inverse problem

Key Result

Lemma 3.1

\newlabellem3.1 Suppose $u^s\in C^4({\mathbb R}^d\setminus\overline\Omega)\cap C^3({\mathbb R}^d\setminus\Omega)$ satisfies $\Delta^2u^s-k^4u^s=0$ in ${\mathbb R}^d\setminus\overline\Omega$ and the radiation condition (1.2), then

Theorems & Definitions (46)

  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • Theorem 3.4
  • proof
  • Lemma 3.5
  • proof
  • Theorem 3.6
  • ...and 36 more