Strong unique continuation for the Lamé system with Lipschitz coefficients in three dimensions
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 3, p. 761-770

This paper studies the strong unique continuation property for the Lamé system of elasticity with variable Lamé coefficients λ, µ in three dimensions, $\mathrm{div}\left(\mu \left(\nabla u+\nabla {u}^{t}\right)\right)+\nabla \left(\lambda \mathrm{div}u\right)+Vu=0$ where λ and μ are Lipschitz continuous and V L∞. The method is based on the Carleman estimate with polynomial weights for the Lamé operator.

DOI : https://doi.org/10.1051/cocv/2010021
Classification:  35B60,  74B05
Keywords: Lamé system, Carleman estimate, strong unique continuation
@article{COCV_2011__17_3_761_0,
author = {Yu, Hang},
title = {Strong unique continuation for the Lam\'e system with Lipschitz coefficients in three dimensions},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {17},
number = {3},
year = {2011},
pages = {761-770},
doi = {10.1051/cocv/2010021},
zbl = {1227.35109},
mrnumber = {2826979},
language = {en},
url = {http://www.numdam.org/item/COCV_2011__17_3_761_0}
}

Yu, Hang. Strong unique continuation for the Lamé system with Lipschitz coefficients in three dimensions. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 3, pp. 761-770. doi : 10.1051/cocv/2010021. http://www.numdam.org/item/COCV_2011__17_3_761_0/

[1] G. Alessandrini and A. Morassi, Strong unique continuation for the Lamé system of elasticity. Comm. P. D. E. 26 (2001) 1787-1810. | MR 1865945 | Zbl 1086.35016

[2] D.D. Ang, M. Ikehata, D.D. Trong and M. Yamamoto, Unique continuation for a stationary isotropic Lamé system with varaiable coefficients. Comm. P. D. E. 23 (1998) 371-385. | MR 1608540 | Zbl 0892.35054

[3] B. Dehman and L. Robbiano, La propriété du prolongement unique pour un système elliptique : le système de Lamé. J. Math. Pures Appl. 72 (1993) 475-492. | MR 1239100 | Zbl 0832.73012

[4] M. Eller, Carleman estimates for some elliptic systems. J. Phys. Conference Series 124 (2008) 012023.

[5] L. Escauriaza, Unique continuation for the system of elasticity in the plan. Proc. Amer. Math. Soc. 134 (2005) 2015-2018. | MR 2215770 | Zbl 1158.35353

[6] C.E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data. Ann. Math. 165 (2007) 567-591. | MR 2299741 | Zbl 1127.35079

[7] C.-L. Lin, G. Nakamura and J.-N. Wang, Optimal three-ball inequalities and quantitative uniqueness for the Lamé system with Lipschitz coefficients. arXiv:0901.4638 (2009). | MR 2730376 | Zbl 1202.35325

[8] C.-L. Lin and J.-N. Wang, Strong unique continuation for the Lamé system with Lipschitz coefficients. Math. Ann. 331 (2005) 611-629. | MR 2122542 | Zbl 1082.35041

[9] A. Martinez, An introduction to semiclassical and microlocal analysis. Springer-Verlag (2002). | MR 1872698 | Zbl 0994.35003

[10] R. Regbaoui, Strong uniqueness for second order differential operators J. Differ. Equ. 141 (1997) 201-217. | MR 1488350 | Zbl 0887.35040

[11] M. Salo and L. Tzou, Carleman estimates and inverse problems for Dirac operators. Math. Ann. 344 (2009) 161-184. | MR 2481057 | Zbl 1169.35063

[12] N. Weck, Außnraumaufgaben in der Theorie stationärer Schwingungen inhomogener elasticher Körper. Math. Z. 111 (1969) 387-398. | MR 263295 | Zbl 0176.09202

[13] N. Weck, Unique continuation for systems with Lamé principal part. Math. Methods Appl. Sci. 24 (2001) 595-605. | MR 1834916 | Zbl 0986.35117

[14] H. Yu, Three spheres inequalities and unique continuation for a three-dimensional Lamé system of elasticity with C1 coeffients. arXiv:0811.1262 (2008).