Quantitative uniqueness estimates for the shallow shell system and their application to an inverse problem

Di Cristo, Michele; Lin, Ching-Lung; Wang, Jenn-Nan

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 1, pp. 43-92.

In this paper we derive some quantitative uniqueness estimates for the shallow shell equations. Our proof relies on appropriate Carleman estimates. For applications, we consider the size estimate inverse problem.

Classification : 53J58, 35R30

[1] G. Alessandrini and A. Morassi, Strong unique continuation for the Lamé system of elasticity, Comm. Partial Differential Equations 26 (2001), 1787–1810. | MR | Zbl

[2] G. Alessandrini, A. Morassi and E. Rosset, Detecting an inclusion in an elastic body by boundary measurements, SIAM J. Math. Anal. 33 (2002), 1247–1268. | MR | Zbl

[3] G. Alessandrini, A. Morassi and E. Rosset, Size estimates, In: “Inverse Problems: Theory and Applications” (Cortona/Pisa, 2002), Contemp. Math., Vol. 333, Amer. Math. Soc., Providence, RI, 2003, 1–33. | MR | Zbl

[4] G. Alessandrini, A. Morassi, E. Rosset and S. Vessella, On doubling inequalities for elliptic systems, J. Math. Anal. Appl. 357 (2009), 349–355. | MR | Zbl

[5] G. Alessandrini, A. Morassi and E. Rosset, The linear constraints in Poincaré and Korn type inequalities, Forum Math. 20 (2008), 557–569. | MR | Zbl

[6] G. Alessandrini and E. Rosset, The inverse conductivity problem with one measurement: bounds on the size of the unknown object, SIAM J. Appl. Math. 58 (1998), 1060–1071. | MR | Zbl

[7] G. Alessandrini, E. Rosset and J. K. Seo, Optimal size estimates for the inverse conductivity problem with one measurement, Proc. Amer. Mat. Soc. 128 (2000), 53–64. | MR | Zbl

[8] J. Bourgain and C. Kenig, On localization in the continuous Anderson-Bernoulli model in higher dimension, Invent. Math. 161 (2005), 1389–426. | MR | Zbl

[9] H. Donnelly and C. Fefferman, Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math. 93 (1988), 161–183. | EuDML | MR | Zbl

[10] N. Garofalo and F. H. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J. 35 (1986), 245–267. | MR | Zbl

[11] N. Garofalo and F. H. Lin, Unique continuation for elliptic operators: a geometric-variational approach, Comm. Pure Appl. Math., 40 (1987), 347–366. | MR | Zbl

[12] L. Hörmander, “The Analysis of Linear Partial Differential Operators”, Vol. 3, Springer-Verlag, Berlin/New York, 1985. | MR | Zbl

[13] V. Isakov, “Inverse Problems for Partial Differential Equations”, Second Edition, Springer, New York, 2006. | MR | Zbl

[14] I. Kukavica, Quantitative uniqueness for second order elliptic operators, Duke Math. J. 91 (1998), 225–240. | MR | Zbl

[15] C.L. Lin, Strong unique continuation for $m$-th powers of a Laplaceian operator with singular coefficients, Proc. Amer. Math. Soc. 135 (2007), 569–578. | MR | Zbl

[16] F. H. Lin, A uniqueness theorem for parabolic equations, Comm. Pure Appl. Math. 43 (1990), 127–136. | MR | Zbl

[17] C. L. Lin, S. Nagayasu and J. N. Wang, Quantitative uniqueness for the power of the Laplacian with singular coefficients, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10 (2011), 513–529 | Numdam | MR | Zbl

[18] C. L. Lin, G. Nakamura and J. N. Wang, Quantitative uniqueness for second order elliptic operators with strongly singular coefficients, preprint. | MR | Zbl

[19] 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 [math.AP] | MR | Zbl

[20] A. Morassi, E. Rosset and S. Vessella, Size estimates for inclusions in an elastic plate by boundary measurements, Indiana Univ. Math. J. 56 (2007), 2325–2384. | MR | Zbl

[21] A. Morassi, E. Rosset and S. Vessella, Detecting general inclusions in elastic plates, Inverse Problems 25 (2009). | MR | Zbl