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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Recovery of a source term or a speed with one measurement and applications


Authors: Plamen Stefanov and Gunther Uhlmann
Journal: Trans. Amer. Math. Soc. 365 (2013), 5737-5758
MSC (2010): Primary 35L05, 35R30
Published electronically: April 25, 2013
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Abstract: We study the problem of recovery of the source $ a(t,x)F(x)$ in the wave equation in anisotropic medium with $ a$ known so that $ a(0,x)\not =0$, with a single measurement. We use Carleman estimates combined with geometric arguments and give sharp conditions for uniqueness. We also study the non-linear problem of recovery of the sound speed in the equation $ u_{tt} -c^2(x)\Delta u =0$ with one measurement. We give sharp conditions for stability as well. An application to thermoacoustic tomography is also presented.


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Additional Information

Plamen Stefanov
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Gunther Uhlmann
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05703-0
PII: S 0002-9947(2013)05703-0
Received by editor(s): March 13, 2011
Received by editor(s) in revised form: August 10, 2011
Published electronically: April 25, 2013
Additional Notes: The first author was partially supported by an NSF Grant DMS-0800428 and a Simons Visiting Professorship
The second author was partially supported by an NSF, a Senior Clay Award and Chancellor Professorship at the University of California, Berkeley
Article copyright: © Copyright 2013 American Mathematical Society