Boundary rigidity with partial data

Abstract

We study the boundary rigidity problem with partial data consisting of determining locally the Riemannian metric of a Riemannian manifold with boundary from the distance function measured at pairs of points near a fixed point on the boundary. We show that one can recover uniquely and in a stable way a conformal factor near a strictly convex point where we have the information. In particular, this implies that we can determine locally the isotropic sound speed of a medium by measuring the travel times of waves joining points close to a convex point on the boundary.The local results lead to a global lens rigidity uniqueness and stability result assuming that the manifold is foliated by strictly convex hypersurfaces. 1 arXiv:1306.2995v5 [math.DG] 8 Oct 2015 2. Reducing the non-linear problem to a pseudo-linear one

Recovery of Material Parameters in Transversely Isotropic Media

“…The main terms in (3.2) are the jl i (x, ξ) terms; the others can be absorbed into these by Poincaré inequalities, at least if the jl i (x, ξ) terms are non-degenerate, see [19]. To leading order at the boundary these decouple due to the δ j i , so one is essentially working on a microlocally weighted X-ray transform combining the differences of the unknown material parameters; more precisely one has a transform for each derivative of the combinations of the differences of these unknown material parameters.…”

“…To leading order at the boundary these decouple due to the δ j i , so one is essentially working on a microlocally weighted X-ray transform combining the differences of the unknown material parameters; more precisely one has a transform for each derivative of the combinations of the differences of these unknown material parameters. (One of course has to deal with these transforms together as done in [19] and follow-up papers.) Thus, one may consider the simplified transforms Roughly speaking, the hypothesis of this proposition says that ignoring coupling one can recover the derivatives of f l (due to ellipticity, choosing the artificial boundary sufficiently close to ∂M ), which then, as the conclusion states, allows one to recover f l , though due to the coupling in LJ, a Poincaré inequality based argument is needed as in [19, Corollary 3.1].…”

“…At the artificial boundary a bit more care is required, and it requires an explicit discussion of scattering covectors and maps related to them. Since elsewhere in the paper only the statement of the lemma is used, we do not recall the background here in more detail, but see for instance [23], [19]. The argument presented below is a modification, keeping track of potential degeneracies in identifications, of the arguments discussed above for the case of points away from the artificial boundary.…”

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

“…We need to extend the geometrical optics solution (27) and reflect it at the boundary to satisfy initial conditions. We consider the phase function = x n = d x z 0 that is globally defined by simplicity assumption.…”

The Light Ray Transform on Lorentzian Manifolds

“…Locally, every Lorentzian metric can be put in this form,. As shown in [41], the Dirichlet-to-Neumann map Λ for the wave operator g + q is an FIO of order zero away from the diagonal, with the canonical relation equal to the lens relation L associated with g. In particular, we recover L. The linearization of L near a fixed g is a light ray transform but it involves derivatives of the perturbation δg , see, for example, [40] for the time independent case. Instead of linearizing L, we will linearize the travel times between boundary points, defined locally as we explain below.…”