The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary

By Matti Lassas, Michael Taylor, Gunther Uhlmann. 2003

185 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Electrical and Electronic Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f487efc4a0829a81da1b0d7f576206d3e33324aa

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Electrical and Electronic Engineering

In Mathematical Physics

Citations in the corpus

Inverse problems: seeing the unseen (2014, background): “) The first constructions in this direction were based on blowing up the metric around a point [125].”
Electrical impedance tomography and Calderón's problem (2009, background): “) The first constructions in this direction were based on blowing up the metric around a point [84].”
The Calderón problem in transversally anisotropic geometries (2013, background): “This statement has only been proved for real-analytic metrics [26] with topological assumptions relaxed in [24], [25], and for Einstein metrics (which are real-analytic in the interior) [12].”
Fractional anisotropic Calderón problem on closed Riemannian manifolds (2021, background): “In dimensions n ≥ 3, it is solved for real-analytic manifolds in [27], [28], [29], see also [19].”
Upper bounds for Steklov eigenvalues on surfaces (2012, background): “The Dirichlet-to-Neumann map has important applications to inverse problems [6, 19].”
Calderon inverse Problem with partial data on Riemann Surfaces (2009, background): “…for instance the surface case by Lassas-Uhlmann [21] (see also [4, 15]), the real-analytic manifold case by Lassas-Taylor-Uhlmann [20] (see also [14] for the Einstein case), the case of manifolds admitting limiting Carleman weights and in a same conformal class by Dos Santos…”
Calderóns' Inverse Problem for Anisotropic Conductivity in the Plane (2004, methodology): “For Riemannian manifolds this kind of technique has been generalized in Lee and Uhlmann (1989), Lassas and Uhlmann (2001), Lassas et al. (2003) .”
Recent progress in the boundary control method (2007, background): “The only known result [81]44 concerns a real analytic case but such analyticity is not natural in this kind of IP.”
Full-Wave Invisibility of Active Devices at All Frequencies (2006, methodology): “A related example of a complete but noncompact two-dimensional Riemannian manifold with boundary having the same Dirichlet-to-Neumann map as a compact one was given in [ LTU ].”
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.”
Limiting Carleman weights and anisotropic inverse problems (2008, methodology): “The result of [17] in dimension n ≥ 3 was extended by Lassas-Taylor-Uhlmann [16] to non-compact, connected real-analytic manifolds with boundary.”
REGULARIZED D-BAR METHOD FOR THE INVERSE CONDUCTIVITY PROBLEM (2009, background): “Counterexamples can be found when the conductivity is allowed to be only positive semi-definite [30, 31, 61].”