Stable determination of a simple metric, a covector field and a potential from the hyperbolic Dirichlet-to-Neumann map

TL;DR

This paper establishes stability estimates for recovering the metric, covector field, and potential of a hyperbolic PDE on a Riemannian manifold from boundary measurements, advancing inverse problem theory.

Contribution

It provides the first stability results for simultaneous determination of the metric, covector field, and potential from the hyperbolic Dirichlet-to-Neumann map.

Findings

Proves Hölder stability estimates near generic simple metrics.

Demonstrates unique recovery of g, b, and q from boundary data.

Extends inverse boundary value problem results to more general geometric settings.

Abstract

Let (M,g) be a compact Riemmanian manifold with non-empty boundary. Consider the second order hyperbolic initial-boundary value problem (\delta_t^2 + P(x,D))u = 0 in (0,T) x M, u(0,x) = \delta_t u(0,x) = 0 for x in M, u(t,x) = f(t,x) on (0,T) x \delta M; where P(x,D) is a first-order perturbation of the Laplace-Beltrami operator on (M,g). Let b and q be the covector field and the potential of P(x,D), respectively, in M. We prove H\"older type stability estimates near generic simple Riemannian metrics for the inverse problem of recovering g, b, and q from the hyperbolic Dirichlet-to-Neumann(DN) map associated, f maps to <\nu,u>_g - i<\nu,b>_g u|_{\delta M x [0,T]} where v is unit conormal to the boundary, modulo a class of transformations that fixed the DN map.