Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Determination of the density in a nonlinear elastic wave equation

Gunther Uhlmann, Jian Zhai

Abstract

This is a continuation of our study [Uhlmann-Zhai, JMPA, 2021] on an inverse boundary value problem for a nonlinear elastic wave equation. We prove that all the linear and nonlinear coefficients can be recovered from the displacement-to-traction map, including the density, under some natural geometric conditions on the wavespeeds.

Determination of the density in a nonlinear elastic wave equation

Abstract

This is a continuation of our study [Uhlmann-Zhai, JMPA, 2021] on an inverse boundary value problem for a nonlinear elastic wave equation. We prove that all the linear and nonlinear coefficients can be recovered from the displacement-to-traction map, including the density, under some natural geometric conditions on the wavespeeds.
Paper Structure (18 sections, 3 theorems, 176 equations)

This paper contains 18 sections, 3 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. Gaussian beams
  3. Fermi coordinates
  4. Gaussian beams
  5. S-waves
  6. P-waves
  7. Gaussian beams with reflections
  8. P- incident waves
  9. S- incident waves
  10. Full Gaussian beam asymptotic solutions
  11. Proof of the main theorem
  12. Second order linearization of the displacement-to-traction map
  13. Uniqueness of the wavespeeds
  14. An integral identity
  15. Construction of S-S-P waves
  16. ...and 3 more sections

Key Result

Theorem 1

Assume $\partial\Omega$ is strictly convex with respect to $g_{P/S}$, and either of the following conditions holds If $T>2\,\max\{\mathrm{diam}_S(\Omega),\mathrm{diam}_P(\Omega)\}$, then $\Lambda$ determines $\lambda,\mu,\rho,\mathscr{A},\mathscr{B},\mathscr{C}$ in $\overline{\Omega}$ uniquely.

Theorems & Definitions (4)

  • Theorem 1
  • Lemma 1
  • proof
  • Proposition 1