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An inverse hyperbolic obstacle problem

Mourad Choulli, Hiroshi Takase

TL;DR

The paper addresses the inverse obstacle problem for hyperbolic equations: reconstructing an unknown boundary source on the obstacle boundary from boundary measurements taken on a surrounding domain. It develops a hyperbolic Carleman inequality with tangential boundary terms and combines it with energy estimates to obtain a Hölder-stable reconstruction, formalized as $\|f\|_{H^1(\Gamma\times(0,\tau))}\le C\big(\mathcal{D}_0(f)+\mathcal{D}(f)^{1-\theta}\mathcal{D}_0(f)^{\theta}\big)$ for some $\theta\in(0,1)$. The work also treats the case where the temporal factor $\mathfrak{b}$ is known, yielding a bound on $\|\mathfrak{a}\|_{H^1(\Gamma)}$. A key contribution is the first Carleman estimate of this type for hyperbolic operators with boundary tangential data, together with an explicit Hölder stability result for inverse boundary-source identification from exterior Cauchy data. These results advance the theoretical understanding of inverse source problems in wave propagation with obstacles and have potential implications for noninvasive boundary identification in heterogeneous media.

Abstract

We establish Hölder stability of an inverse hyperbolic obstacle problem. Mainly, we study the problem of reconstructing an unknown function defined on the boundary of the obstacle from two measurements taken on the boundary of a domain surrounding the obstacle.

An inverse hyperbolic obstacle problem

TL;DR

The paper addresses the inverse obstacle problem for hyperbolic equations: reconstructing an unknown boundary source on the obstacle boundary from boundary measurements taken on a surrounding domain. It develops a hyperbolic Carleman inequality with tangential boundary terms and combines it with energy estimates to obtain a Hölder-stable reconstruction, formalized as for some . The work also treats the case where the temporal factor is known, yielding a bound on . A key contribution is the first Carleman estimate of this type for hyperbolic operators with boundary tangential data, together with an explicit Hölder stability result for inverse boundary-source identification from exterior Cauchy data. These results advance the theoretical understanding of inverse source problems in wave propagation with obstacles and have potential implications for noninvasive boundary identification in heterogeneous media.

Abstract

We establish Hölder stability of an inverse hyperbolic obstacle problem. Mainly, we study the problem of reconstructing an unknown function defined on the boundary of the obstacle from two measurements taken on the boundary of a domain surrounding the obstacle.
Paper Structure (7 sections, 4 theorems, 102 equations)

This paper contains 7 sections, 4 theorems, 102 equations.

Table of Contents

  1. Introduction and main result
  2. Proof of Theorem \ref{['exHolder']}
  3. Preliminaries
  4. Proof of Theorem \ref{['exHolder']}
  5. Proof of Proposition \ref{['global_Carleman_estimate']}
  6. Analysis of the IBVP
  7. Proof of Proposition \ref{['energy_estimate']}

Key Result

Theorem 1.2

Assume that $T>T^\ast$. Let $\tau \in (0,T-T^\ast)$ and $\zeta=(g,\phi,p,T,\tau,\beta/\alpha)$. Then there exist constants $C=C(\zeta)>0$ and $\theta=\theta(\zeta)\in(0,1)$ such that for any $f\in\mathscr{F}$ we have

Theorems & Definitions (9)

  • Example 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • proof
  • proof