Limiting Carleman weights and anisotropic inverse problems

By David dos Santos Ferreira, Carlos E. Kenig, Mikko Salo, Gunther Uhlmann. 2008

292 citations recorded by the source corpus.

In this article we consider the anisotropic Calderón problem and related inverse problems. The approach is based on limiting Carleman weights, introduced in Kenig et al. (Ann. Math. 165:567–591, 2007) in the Euclidean case. We characterize those Riemannian manifolds which admit limiting Carleman weights, and give a complex geometrical optics construction for a class of such manifolds. This is used to prove uniqueness results for anisotropic inverse problems, via the attenuated geodesic ray transform. Earlier results in dimension n≥3 were restricted to real-analytic metrics.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7f2a62ef65430c69d34c3cc5e6d206dc30a12085

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Citations in the corpus

Determining an Unbounded Potential from Cauchy Data in Admissible Geometries (2011, methodology): “In [4] anisotropic inverse problems were considered in certain admissible geometries, that is, on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold.”
The Calderón problem with partial data in two dimensions (2010, background): “The latter fact follows from the unique boundary determination of potentials from the Dirichlet-to-Neumann map (see for instance [12], [29]).”
The Calderón problem for variable coefficients nonlocal elliptic operators (2017, methodology): “Note that we do not assume any further special structures on A(x) unlike to the case s = 1, for example, the method [17] consists of considering the limiting Carleman weight function for the Laplace–Beltrami operator inM and constructing the corresponding complex geometrical optics (CGO) solutions b”
Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets (2012, result): “This problem remains open, however, and we refer to [15, 20, 34] for partial results.”
The Calderón problem with partial data on manifolds and applications (2012, methodology): “In the following we will use some general facts on limiting Carleman weights from [Ferreira 2009a], where it was proved that any limiting Carleman weight in R3 has, up to translation, rotation and scaling, one of the following six forms:”
The attenuated ray transform on simple surfaces (2010, methodology): “Rather unexpectedly, this transform also appeared in the recent works [7], [13] in the context of Calderón’s inverse conductivity problem in anisotropic media.”
Inverse problems: seeing the unseen (2014, methodology): “In [52] we have classified locally all the LCW in Euclidean space.”
Fractional anisotropic Calderón problem on closed Riemannian manifolds (2021, background): “In the C case, the uniqueness for the anisotropic Calderón problem has only been established in [10], [11] for metrics in a fixed conformal class in the case of a conformally transversally anisotropic manifold 1”
Lens rigidity for manifolds with hyperbolic trapped set (2014, unknown): “by R(x,v)w:= R(w,v)v∈ E(x,v), and X acts on sections of Eby diﬀerentiating parallel transport along the geodesic (see Section 2 of [PSU]). Then the proof of Lemma 11.2 of [PSU] and Proposition 7.2 of [DKSU] is based on Santalo’s formula (2.20) and thus applies as well in our setting (ie. the boundar”
The Calderón inverse problem for isotropic quasilinear conductivities (2021, background): “Let us remark that CGO solutions were introduced in [5] for the case γ0 = 1 and developed further for arbitrary positive γ0 in [45], see [12] for generalization of CGO solutions in the context of certain classes of Riemannian manifolds.”
The Calderón problem in transversally anisotropic geometries (2013, methodology): “The complex geometrical optics solutions in [8], satisfying (−∆g + q)u = 0 in M , have the form”
On Lp resolvent estimates for Laplace–Beltrami operators on compact manifolds (2011, methodology): “This work grew out of our recent interest in L Carleman estimates with limiting Carleman weights generalizing Jerison and Kenig’s estimates [11] to variable coefficients and used in the context of anisotropic inverse problems [4,5].”