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Asymptotic behavior of the indicator function in the inverse problem of the wave equation for media with multiple types of cavities

Mishio Kawashita, Wakako Kawashita

Abstract

In this paper, the inverse problem of the wave equation by the enclosure method for a medium with multiple types of cavities is discussed. In the case considered here, the sign of the indicator function of the enclosure method is not determined and sign cancellation may occur, resulting in loss of information. By examining the top terms of the indicator function in detail, we show that the shortest distance to the cavities can be obtained even in such a case.

Asymptotic behavior of the indicator function in the inverse problem of the wave equation for media with multiple types of cavities

Abstract

In this paper, the inverse problem of the wave equation by the enclosure method for a medium with multiple types of cavities is discussed. In the case considered here, the sign of the indicator function of the enclosure method is not determined and sign cancellation may occur, resulting in loss of information. By examining the top terms of the indicator function in detail, we show that the shortest distance to the cavities can be obtained even in such a case.
Paper Structure (12 sections, 13 theorems, 164 equations, 2 figures)

This paper contains 12 sections, 13 theorems, 164 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Non-degenerate condition
  3. Main results and a reduction of $I_{\tau}$
  4. Construction of an approximate solution
  5. Asymptotic form of the main terms of $I_{\tau}$
  6. Estimates of the remainder terms
  7. Estimate of $g_{\tau, N}$
  8. Estimate of $f_{\tau, N}$
  9. Estimates of $h_{\tau}$
  10. Estimates of $J_{\tau, N}^{\alpha,r}$
  11. Appendix.
  12. Estimates of weak solutions

Key Result

Theorem 1.2

(M. and W. Kawashita separatedM. and W. Kawashita combined) If all of $D^{n_\pm}$, $D^{d}$ and $B$ have $C^1$ boundaries, the indicator function (Indicator function for Cave&Inclusion) satisfies the following properties: (1) For $T < 2l_0/\sqrt{\gamma_0}$, $\lim_{\tau \to \infty}e^{{\tau}T}I_\tau = Further, in each of the cases $(ii)_+$ and $(ii)_{-}$, we also have

Figures (2)

  • Figure 1: Example \ref{['lambda1=0']}
  • Figure 2: Example \ref{['lambda1not=0']}

Theorems & Definitions (21)

  • Remark 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • Example 3.4
  • Example 3.5
  • ...and 11 more