Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map

Mourad Bellassoued; David Dos Santos Ferreira

doi:10.3934/ipi.2011.5.745

Article Contents

2011, Volume 5, Issue 4: 745-773. Doi: 10.3934/ipi.2011.5.745

This issue Previous Article Next Article Recovery of the heat coefficient by two measurements

Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map

Mourad Bellassoued^1, and
David Dos Santos Ferreira^2,

1.
University of Carthage, Department of Mathematics, Faculty of Sciences of Bizerte, 7021 Jarzouna Bizerte
2.
Université Paris 13, CNRS, UMR 7539 LAGA, 99, avenue Jean-Baptiste Clément, F-93 430 Villetaneuse

Received: January 31, 2011

Revised: August 31, 2011

Published: October 2011

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Abstract

In this article we seek stability estimates in the inverse problem of determining the potential or the velocity in a wave equation in an anisotropic medium from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the wave equation. We prove in dimension $n\geq 2$ that the knowledge of the Dirichlet-to-Neumann map for the wave equation uniquely determines the electric potential and we prove Hölder-type stability in determining the potential. We prove similar results for the determination of velocities close to 1.

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Mathematics Subject Classification: Primary: 58J32, 35L05; Secondary: 58J50.

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