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15 April 2014 Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets
Matti Lassas, Lauri Oksanen
Duke Math. J. 163(6): 1071-1103 (15 April 2014). DOI: 10.1215/00127094-2649534

Abstract

We consider the inverse problem to determine a smooth compact Riemannian manifold with boundary (M,g) from a restriction ΛS,R of the Dirichlet-to-Neumann operator for the wave equation on the manifold. Here S and R are open sets in M and the restriction ΛS,R corresponds to the case where the Dirichlet data is supported on R+×S and the Neumann data is measured on R+×R. In the novel case where SR=, we show that ΛS,R determines the manifold (M,g) uniquely, assuming that the wave equation is exactly controllable from the set of sources S. Moreover, we show that the exact controllability can be replaced by the Hassell–Tao condition for eigenvalues and eigenfunctions, that is,

λjCνϕjL2(S)2,j=1,2,,

where λj are the Dirichlet eigenvalues and where (ϕj)j=1 is an orthonormal basis of the corresponding eigenfunctions.

Citation

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Matti Lassas. Lauri Oksanen. "Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets." Duke Math. J. 163 (6) 1071 - 1103, 15 April 2014. https://doi.org/10.1215/00127094-2649534

Information

Published: 15 April 2014
First available in Project Euclid: 11 April 2014

zbMATH: 1375.35634
MathSciNet: MR3192525
Digital Object Identifier: 10.1215/00127094-2649534

Subjects:
Primary: 35R30
Secondary: 35R01

Rights: Copyright © 2014 Duke University Press

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Vol.163 • No. 6 • 15 April 2014
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