Uniqueness in an inverse problem of fractional elasticity

TL;DR

This paper investigates the unique determination of material parameters in a fractional elasticity model using boundary measurements, extending classical methods to nonlocal operators with implications for nonlocal elasticity theory.

Contribution

It introduces a fractional matrix Schr"odinger approach and proves uniqueness of Lamé parameters under specific conditions, advancing inverse problems in nonlocal elasticity.

Findings

Unique recovery of Lamé parameters when they are constant outside the domain.

Poisson ratios must agree everywhere for uniqueness.

Extension of classical inverse methods to fractional elasticity context.

Abstract

We study an inverse problem for fractional elasticity. In analogy to the classical problem of linear elasticity, we consider the unique recovery of the Lam\'e parameters associated to a linear, isotropic fractional elasticity operator from fractional Dirichlet-to-Neumann data. In our analysis we make use of a fractional matrix Schr\"odinger equation via a generalization of the so-called Liouville reduction, a technique classically used in the study of the scalar conductivity equation. We conclude that unique recovery is possible if the Lam\'e parameters agree and are constant in the exterior, and their Poisson ratios agree everywhere. Our study is motivated by the significant recent activity in the field of nonlocal elasticity.