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Strong unique continuation for the Lamé system with Lipschitz coefficients in three dimensions*

Published online by Cambridge University Press:  23 April 2010

Hang Yu
Hang Yu*
Affiliation:
School of Mathematical Sciences, Fudan University, 200433 Shanghai, P.R. China. hangyumath@hotmail.com
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Abstract

This paper studies the strong unique continuation property for the Lamé system of elasticity with variable Lamé coefficients λ, µ in three dimensions, ${\rm{div}}(\mu(\nabla u+\nabla u^t))+ \nabla(\lambda{\rm{div}} u)+Vu=0$ 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.

Keywords

Lamé systemCarleman estimatestrong unique continuation
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 3 , July 2011 , pp. 761 - 770
Copyright
© EDP Sciences, SMAI, 2010

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References

Alessandrini, G. and Morassi, A., Strong unique continuation for the Lamé system of elasticity. Comm. P. D. E. 26 (2001) 1787–1810. CrossRef
Ang, D.D., Ikehata, M., Trong, D.D. and Yamamoto, M., Unique continuation for a stationary isotropic Lamé system with varaiable coefficients. Comm. P. D. E. 23 (1998) 371–385. CrossRef
Dehman, B. and Robbiano, L., La propriété du prolongement unique pour un système elliptique : le système de Lamé. J. Math. Pures Appl. 72 (1993) 475–492.
Eller, M., Carleman estimates for some elliptic systems. J. Phys. Conference Series 124 (2008) 012023. CrossRef
Escauriaza, L., Unique continuation for the system of elasticity in the plan. Proc. Amer. Math. Soc. 134 (2005) 2015–2018. CrossRef
Kenig, C.E., Sjöstrand, J. and Uhlmann, G., The Calderón problem with partial data. Ann. Math. 165 (2007) 567–591. CrossRef
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).
Lin, C.-L. and Wang, J.-N., Strong unique continuation for the Lamé system with Lipschitz coefficients. Math. Ann. 331 (2005) 611–629. CrossRef
A. Martinez, An introduction to semiclassical and microlocal analysis. Springer-Verlag (2002).
R. Regbaoui, Strong uniqueness for second order differential operators J. Differ. Equ. 141 (1997) 201–217.
Salo, M. and Tzou, L., Carleman estimates and inverse problems for Dirac operators. Math. Ann. 344 (2009) 161–184. CrossRef
Weck, N., Außnraumaufgaben in der Theorie stationärer Schwingungen inhomogener elasticher Körper. Math. Z. 111 (1969) 387–398. CrossRef
Weck, N., Unique continuation for systems with Lamé principal part. Math. Methods Appl. Sci. 24 (2001) 595–605. CrossRef
H. Yu, Three spheres inequalities and unique continuation for a three-dimensional Lamé system of elasticity with C 1 coeffients. arXiv:0811.1262 (2008).