Stability estimates in tensor tomography

Jan Boman; Vladimir Sharafutdinov

doi:10.3934/ipi.2018052

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2018, Volume 12, Issue 5: 1245-1262. Doi: 10.3934/ipi.2018052

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Stability estimates in tensor tomography

Jan Boman^1, and
Vladimir Sharafutdinov^2,3, ,

1.
Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden
2.
Sobolev Institute of Mathematics, 4 Koptyug Avenue, Novosibirsk, 630090, Russia
3.
Novosibirsk State University, 2 Pirogov street, Novosibirsk, 630090, Russia

* Corresponding author: Vladimir Sharafutdinov
* Corresponding author: Vladimir Sharafutdinov

Received: October 2017

Published: October 2018

The second author was supported by RFBR, Grant 17-51-150001.

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Abstract

We study the X-ray transform $I$ of symmetric tensor fields on a smooth convex bounded domain $Ω\subset{\mathbb R}^n$. The main result is the stability estimate $\|^{s}f\|_{L^2}≤ C\|If\|_{H^{1/2}}$, where $^{s}f$ is the solenoidal part of the tensor field $f$. The proof is based on a comparison of the Dirichlet integrals for the exterior and interior Dirichlet problems and on a generalization of the Korn inequality to symmetric tensor fields of arbitrary rank.

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Mathematics Subject Classification: Primary: 44A15, 35J20; Secondary: 74B05.

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