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Rigidity of homogeneous Lamé systems

Joonas Ilmavirta, Teemu Saksala, Lili Yan

TL;DR

The paper proves a rigidity result for isotropic elastic media: if the hyperbolic Dirichlet-to-Neumann map for a homogeneous Lamé system agrees with that of a possibly inhomogeneous one beyond a threshold time, and the first model is simple with a convex foliation while the second has no geometric assumptions, then the two Lamé triplets coincide. The approach combines boundary determination, boundary-distance data for $p$- and $s$-waves, and distance-rigidity arguments in a simple geometry to show equality of the conformal factors, densities, and eventually the full Lamé coefficients. The novelty is that the second model is unconstrained geometrically, yet the DN-map determines it uniquely given the first model's strong geometric hypotheses. This extends boundary rigidity results into elastic inverse problems by exploiting the two-wave structure and recent convex-foliation-based uniqueness results.

Abstract

In this short paper, we show that any Lamé system whose Dirichlet-to-Neumann map for the elastic wave equation agrees with the one arising from the homogeneous Lamé system must actually be homogeneous. We do not need to impose any assumptions for the Lamé coefficients that we aim to recover. We use the fact that the homogeneous system gives rise to a geometry that is both simple and admits a strictly convex foliation.

Rigidity of homogeneous Lamé systems

TL;DR

The paper proves a rigidity result for isotropic elastic media: if the hyperbolic Dirichlet-to-Neumann map for a homogeneous Lamé system agrees with that of a possibly inhomogeneous one beyond a threshold time, and the first model is simple with a convex foliation while the second has no geometric assumptions, then the two Lamé triplets coincide. The approach combines boundary determination, boundary-distance data for - and -waves, and distance-rigidity arguments in a simple geometry to show equality of the conformal factors, densities, and eventually the full Lamé coefficients. The novelty is that the second model is unconstrained geometrically, yet the DN-map determines it uniquely given the first model's strong geometric hypotheses. This extends boundary rigidity results into elastic inverse problems by exploiting the two-wave structure and recent convex-foliation-based uniqueness results.

Abstract

In this short paper, we show that any Lamé system whose Dirichlet-to-Neumann map for the elastic wave equation agrees with the one arising from the homogeneous Lamé system must actually be homogeneous. We do not need to impose any assumptions for the Lamé coefficients that we aim to recover. We use the fact that the homogeneous system gives rise to a geometry that is both simple and admits a strictly convex foliation.
Paper Structure (2 sections, 7 theorems, 27 equations)

This paper contains 2 sections, 7 theorems, 27 equations.

Table of Contents

  1. Uniqueness of Lamé triplets
  2. Proofs

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{R}^3$ be an open, bounded, and strictly convex domain with smooth boundary. Let $(\lambda_1,\mu_1,\rho_1)$ be constants and let $(\lambda_2,\mu_2,\rho_2)$ smooth functions, both satisfying positivity conditions eq:strong-convexity. If the hyperbolic Dirichlet-to-Neumann m

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1: Novelty of this paper
  • Remark 1.2
  • Lemma 2.1: rachele2000boundary
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • proof : Proof of Lemma \ref{['lem:Rac2']}
  • Remark 2.1
  • ...and 3 more