Rigidity of homogeneous Lam\'e systems

TL;DR

This paper proves that if the Dirichlet-to-Neumann map of an elastic wave system matches that of a homogeneous Lamé system, then the system must be homogeneous, using geometric properties of the associated homogeneous system.

Contribution

It establishes a uniqueness result for the Lamé coefficients based solely on boundary measurements, without additional assumptions.

Findings

Matching Dirichlet-to-Neumann maps imply homogeneity of the Lamé system.

The homogeneous system's geometry admits a strictly convex foliation.

No extra assumptions are needed on the Lamé coefficients.

Abstract

In this short paper, we show that any Lam\'e system whose Dirichlet-to-Neumann map for the elastic wave equation agrees with the one arising from the homogeneous Lam\'e system must actually be homogeneous. We do not need to impose any assumptions for the Lam\'e 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.