Stable Determination of a Simple Metric, a Covector Field and a Potential from the Hyperbolic Dirichlet-to-Neumann Map

Abstract

Letg be a compact Riemmanian manifold with non-empty boundary. Consider the second order hyperbolic initial-boundary value problemis a first-order perturbation of the Laplace operator − g on g . Here b and q are a covector field and a potential, respectively, in . We prove Hölder type stability estimates near generic simple Riemannian metrics for the inverse problem of recovering simultaneously g, b, and q from the hyperbolic Dirichletto-Neumann (DN) map associated, f → u − i b g u × 0 T modulo a class of transformations that fixed the hyperbolic DN map.

The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds

“…We are looking for an amplitude a of the form a ∼ ∞ j=0 a j (x, ξ ), where a j is homogeneous in the ξ variable of degree −j. The standard geometric optics construction leads to the transport equations (25), (26). Using the standard Borel lemma argument, we construct a convergent series for a.…”

“…Proof. We adapt the proofs in [26] and [38] in the Riemannian setting. Let Γ 0 be a small conic neighborhood of ξ 0 .…”

“…represents the stably determined terms of order O(δ ). (In fact, a 1 = 0 in these expressions by the boundary condition in(26), but it is left here for the convenience of tracking the corresponding terms.) From the estimates(35) (44)and (47) it follows that…”

The Geodesic Ray Transform on Riemannian Surfaces with Conjugate Points

“…This linear problem is also in the heart of the non-linear problem of recovery a metric or a sound speed (a conformal factor) from the lengths of the geodesics measured at the boundary (boundary rigidity) or from knowledge of the lens relation (lens rigidity), see, e.g., [7,6,8,25,28,36,34,32,29]; or from knowledge of the hyperbolic Dirichlet-to-Neumann (DN) map [3,4,23,27,33,30,2]. It is the linearization of the first two (f is a tensor field then); and the lens relation is directly related to the DN map and its canonical relation as an FIO.…”

“…Understanding the stability of X is therefore fundamental for all those problems. Recovery of lower order terms like a magnetic and an electric fields from the associated DN maps leads to X-ray transforms of those fields along the geodesics [23] as well.…”