Boundary determination of the Riemannian metric from Cauchy data for the Stokes equations

TL;DR

This paper proves that the boundary measurements for the Stokes equations uniquely determine all boundary derivatives of the Riemannian metric on a compact manifold, advancing inverse boundary value problem theory.

Contribution

It establishes the boundary determination of the Riemannian metric's derivatives from Cauchy data for the Stokes equations on a compact manifold.

Findings

Unique determination of boundary derivatives of the metric

Extension of inverse boundary problem results to Stokes equations

Provides a foundation for reconstructing interior metrics from boundary data

Abstract

For a compact connected Riemannian manifold of dimension $n$ with smooth boundary, $n\geqslant 2$, we prove that the Cauchy data (or the Dirichlet-to-Neumann map) for the Stokes equations uniquely determines the partial derivatives of all orders of the metric on the boundary of the manifold.