Reconstructions from boundary measurements on admissible manifolds

Carlos E. Kenig; Mikko Salo; Gunther Uhlmann

doi:10.3934/ipi.2011.5.859

Article Contents

2011, Volume 5, Issue 4: 859-877. Doi: 10.3934/ipi.2011.5.859

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Reconstructions from boundary measurements on admissible manifolds

1.
Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL 60637-1514
2.
Department of Mathematics and Statistics, University of Helsinki and University of Jyväskylä, PO Box 68, 00014 Helsinki
3.
Department of Mathematics, University of Washington and, Department of Mathematics, University of California, Irvine, CA 92697-3875

Received: November 2010

Revised: August 2011

Published: November 2011

Abstract / Introduction Related Papers Cited by

Abstract

We prove that a potential $q$ can be reconstructed from the Dirichlet-to-Neumann map for the Schrödinger operator $-\Delta_g + q$ in a fixed admissible $3$-dimensional Riemannian manifold $(M,g)$. We also show that an admissible metric $g$ in a fixed conformal class can be constructed from the Dirichlet-to-Neumann map for $\Delta_g$. This is a constructive version of earlier uniqueness results by Dos Santos Ferreira et al. [10] on admissible manifolds, and extends the reconstruction procedure of Nachman [31] in Euclidean space. The main points are the derivation of a boundary integral equation characterizing the boundary values of complex geometrical optics solutions, and the development of associated layer potentials adapted to a cylindrical geometry.

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Mathematics Subject Classification: Primary: 35R30; Secondary: 58J32.

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