Duke Mathematical Journal

Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets

Matti Lassas and Lauri Oksanen

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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,


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

Article information

Duke Math. J., Volume 163, Number 6 (2014), 1071-1103.

First available in Project Euclid: 11 April 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35R30: Inverse problems
Secondary: 35R01: Partial differential equations on manifolds [See also 32Wxx, 53Cxx, 58Jxx]


Lassas, Matti; Oksanen, Lauri. Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets. Duke Math. J. 163 (2014), no. 6, 1071--1103. doi:10.1215/00127094-2649534. https://projecteuclid.org/euclid.dmj/1397223296

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