American Institute of Mathematical Sciences

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October  2018, 12(5): 1245-1262. doi: 10.3934/ipi.2018052

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

Received  October 2017 Published  July 2018

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

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.

Keywords: Tensor tomography, X-ray transform, the Dirichlet principle, the Korn inequality.
Mathematics Subject Classification: Primary: 44A15, 35J20; Secondary: 74B05.
Citation: Jan Boman, Vladimir Sharafutdinov. Stability estimates in tensor tomography. Inverse Problems & Imaging, 2018, 12 (5) : 1245-1262. doi: 10.3934/ipi.2018052
References:
[1]

G. Duvaut and J.-L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972.  Google Scholar

[2]

G. Fichera, Existence Theorems in Elasticity, Springer, 1972. Google Scholar

[3]

K. O. Friedrichs, On the boundary value problems of the theory of elasticity and Korn's inequality, Ann. Math., 48 (1947), 441-471.  doi: 10.2307/1969180.  Google Scholar

[4]

J. Gobert, Une inégalité fondamentale de la théorie de l'élasticité, Bull. de la Roy. des Sci. de Liège, 31 (1962), 182-191.   Google Scholar

[5]

L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. 1, Springer, 1983. doi: 10.1007/978-3-642-96750-4.  Google Scholar

[6]

V. A. Kondrat'ev and O. A. Oleinik, Boundary value problems for the system of elasticity theory in unbounded domains. Korn's inequalities, Russian Mathematical Surveys, 43 (1988), 65-120.  doi: 10.1070/RM1988v043n05ABEH001945.  Google Scholar

[7]

A. Korn, Solution générale du problème d'équilibre dans la théorie de l'élasticité dans le cas où les efforts sont donnés à la surface, in Ann. Fac. Sci. Univ. Toulouse, 10 (1908), 165–269. doi: 10.5802/afst.251.  Google Scholar

[8]

W. Lionheart and V. Sharafutdinov, Reconstruction algorithm for the linearized polarization tomography problem with incomplete data, in Imaging Microstructures: Mathematical and Computational Challenges, (eds. H. Ammari and Hyeonbae Kang), Contemporary Mathematics, 494 (2009), 137–159. doi: 10.1090/conm/494/09648.  Google Scholar

[9]

S. G. Mikhlin, Variational Methods in Mathematical Physics, Oxford, Pergamon Press, 1964.  Google Scholar

[10]

F. Natterer, The Mathematics of Computerized Tomography, John Willey & Sons, 1986.  Google Scholar

[11]

L. E. Payne and H. F. Weinberger, On Korn's inequality, Arch. Rat. Mech. Anal., 8 (1961), 89-98.  doi: 10.1007/BF00277432.  Google Scholar

[12]

L. N. Pestov and V. A. Sharafutdinov, Integral geometry of tensor fields on a manifold of negative curvature, Siberian Math. J., 29 (1988), 427-441.  doi: 10.1007/BF00969652.  Google Scholar

[13]

Yu. G. Reshetnyak, Estimates for certain differential operators with finite-dimensional kernel, Siberian Math. J., 11 (1970), 315-326.   Google Scholar

[14]

V. A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, The Netherlands, 1994. doi: 10.1515/9783110900095.  Google Scholar

[15]

V. Sharafutdinov, The Reshetnyak formula and Natterer stability estimates in tensor tomography, Inverse Problems, 33 (2017), 025002 (20 pp). doi: 10.1088/1361-6420/33/2/025002.  Google Scholar

[16]

V. Sharafutdinov, M. Skokan and G. Uhlmann, Regularity of ghosts in tensor tomography, J. Geom. Anal., 15 (2005), 499-542.  doi: 10.1007/BF02930983.  Google Scholar

[17]

V. Sharafutdinov and J. Wang J, Tomography of small residual stresses, Inverse Problems, 28 (2012), 065017, 17 pp. doi: 10.1088/0266-5611/28/6/065017.  Google Scholar

[18]

P. Stefanov, A sharp stability estimate in tensor tomography, in J. of Physics: Conference Series, 124 (2008), 012007. doi: 10.1088/1742-6596/124/1/012007.  Google Scholar

[19]

M. E. Taylor, Partial Differential Equations I. Basic Theory, Texts in Applied Mathematics, 23. Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4684-9320-7.  Google Scholar

show all references

References:
[1]

G. Duvaut and J.-L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972.  Google Scholar

[2]

G. Fichera, Existence Theorems in Elasticity, Springer, 1972. Google Scholar

[3]

K. O. Friedrichs, On the boundary value problems of the theory of elasticity and Korn's inequality, Ann. Math., 48 (1947), 441-471.  doi: 10.2307/1969180.  Google Scholar

[4]

J. Gobert, Une inégalité fondamentale de la théorie de l'élasticité, Bull. de la Roy. des Sci. de Liège, 31 (1962), 182-191.   Google Scholar

[5]

L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. 1, Springer, 1983. doi: 10.1007/978-3-642-96750-4.  Google Scholar

[6]

V. A. Kondrat'ev and O. A. Oleinik, Boundary value problems for the system of elasticity theory in unbounded domains. Korn's inequalities, Russian Mathematical Surveys, 43 (1988), 65-120.  doi: 10.1070/RM1988v043n05ABEH001945.  Google Scholar

[7]

A. Korn, Solution générale du problème d'équilibre dans la théorie de l'élasticité dans le cas où les efforts sont donnés à la surface, in Ann. Fac. Sci. Univ. Toulouse, 10 (1908), 165–269. doi: 10.5802/afst.251.  Google Scholar

[8]

W. Lionheart and V. Sharafutdinov, Reconstruction algorithm for the linearized polarization tomography problem with incomplete data, in Imaging Microstructures: Mathematical and Computational Challenges, (eds. H. Ammari and Hyeonbae Kang), Contemporary Mathematics, 494 (2009), 137–159. doi: 10.1090/conm/494/09648.  Google Scholar

[9]

S. G. Mikhlin, Variational Methods in Mathematical Physics, Oxford, Pergamon Press, 1964.  Google Scholar

[10]

F. Natterer, The Mathematics of Computerized Tomography, John Willey & Sons, 1986.  Google Scholar

[11]

L. E. Payne and H. F. Weinberger, On Korn's inequality, Arch. Rat. Mech. Anal., 8 (1961), 89-98.  doi: 10.1007/BF00277432.  Google Scholar

[12]

L. N. Pestov and V. A. Sharafutdinov, Integral geometry of tensor fields on a manifold of negative curvature, Siberian Math. J., 29 (1988), 427-441.  doi: 10.1007/BF00969652.  Google Scholar

[13]

Yu. G. Reshetnyak, Estimates for certain differential operators with finite-dimensional kernel, Siberian Math. J., 11 (1970), 315-326.   Google Scholar

[14]

V. A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, The Netherlands, 1994. doi: 10.1515/9783110900095.  Google Scholar

[15]

V. Sharafutdinov, The Reshetnyak formula and Natterer stability estimates in tensor tomography, Inverse Problems, 33 (2017), 025002 (20 pp). doi: 10.1088/1361-6420/33/2/025002.  Google Scholar

[16]

V. Sharafutdinov, M. Skokan and G. Uhlmann, Regularity of ghosts in tensor tomography, J. Geom. Anal., 15 (2005), 499-542.  doi: 10.1007/BF02930983.  Google Scholar

[17]

V. Sharafutdinov and J. Wang J, Tomography of small residual stresses, Inverse Problems, 28 (2012), 065017, 17 pp. doi: 10.1088/0266-5611/28/6/065017.  Google Scholar

[18]

P. Stefanov, A sharp stability estimate in tensor tomography, in J. of Physics: Conference Series, 124 (2008), 012007. doi: 10.1088/1742-6596/124/1/012007.  Google Scholar

[19]

M. E. Taylor, Partial Differential Equations I. Basic Theory, Texts in Applied Mathematics, 23. Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4684-9320-7.  Google Scholar

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