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RECOVERY OF NON-COMPACTLY SUPPORTED COEFFICIENTS OF ELLIPTIC EQUATIONS ON AN INFINITE WAVEGUIDE

Part of: Miscellaneous topics - Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  05 November 2018

Yavar Kian
Yavar Kian*
Affiliation:
Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France ([email protected])
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Abstract

We consider the unique recovery of a non-compactly supported and non-periodic perturbation of a Schrödinger operator in an unbounded cylindrical domain, also called waveguide, from boundary measurements. More precisely, we prove recovery of a general class of electric potentials from the partial Dirichlet-to-Neumann map, where the Dirichlet data is supported on slightly more than half of the boundary and the Neumann data is taken on the other half of the boundary. We apply this result in different contexts including recovery of some general class of non-compactly supported coefficients from measurements on a bounded subset and recovery of an electric potential, supported on an unbounded cylinder, of a Schrödinger operator in a slab.

Keywords

inverse problemselliptic equationsscalar potentialunbounded domaininfinite cylindrical waveguideslabpartial dataCarleman estimate

MSC classification

Primary: 35R30: Inverse problems 35J15: Second-order elliptic equations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 5 , September 2020 , pp. 1573 - 1600
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Copyright
© Cambridge University Press 2018

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References

Ahluwalia, D. and Keller, J., Exact and Asymptotic Representations of the Sound Field in a Stratified Ocean. Wave Propagation and Underwater Acoustics, Lecture Notes in Physics, vol. 70, pp. 14–85 (Springer, Berlin, 1977).Google Scholar
Ammari, H. and Uhlmann, G., Reconstuction from partial Cauchy data for the Schrödinger equation, Indiana Univ. Math. J. 53 (2004), 169–184.Google Scholar
Bellassoued, M., Kian, Y. and Soccorsi, E., An inverse stability result for non compactly supported potentials by one arbitrary lateral Neumann observation, J. Differential Equations 260 (2016), 7535–7562.Google Scholar
Bellassoued, M., Kian, Y. and Soccorsi, E., An inverse problem for the magnetic Schrödinger equation in infinite cylindrical domains, Publ. Res. Inst. Math. Sci. 54 (2018), 679–728.Google Scholar
Bukhgeim, A., Recovering the potential from Cauchy data in two dimensions, J. Inverse Ill-Posed Probl. 16 (2008), 19–34.Google Scholar
Bukhgeim, A. L. and Uhlmann, G., Recovering a potential from partial Cauchy data, Commun. Partial Differential Equations 27(3–4) (2002), 653–668.Google Scholar
Calderón, A. P., On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics, pp. 65–73 (Sociedade Brasileira de Matematica, Rio de Janeiro, 1980).Google Scholar
Caro, P., Dos Santos Ferreira, D. and Ruiz, A., Stability estimates for the Radon transform with restricted data and applications, Adv. Math. 267 (2014), 523–564.Google Scholar
Caro, P., Dos Santos Ferreira, D. and Ruiz, A., Stability estimates for the Calderón problem with partial data, J. Differential Equations 260 (2016), 2457–2489.Google Scholar
Caro, P. and Marinov, K., Stability of inverse problems in an infinite slab with partial data, Comm. Partial Differential Equations 41 (2016), 683–704.Google Scholar
Chang, P.-Y. and Lin, H.-H., Conductance through a single impurity in the metallic zigzag carbon nanotube, Appl. Phys. Lett. 95 (2009), 082104.Google Scholar
Choulli, M., Une introduction aux problèmes inverses elliptiques et paraboliques, Mathématiques et Applications, Volume 65 (Springer, Berlin, 2009).Google Scholar
Choulli, M. and Kian, Y., Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term, J. Math. Pures Appl. (9) 114 (2018), 235–261.Google Scholar
Choulli, M., Kian, Y. and Soccorsi, E., Stable determination of time-dependent scalar potential from boundary measurements in a periodic quantum waveguide, SIAM J. Math. Anal. 47(6) (2015), 4536–4558.Google Scholar
Choulli, M., Kian, Y. and Soccorsi, E., Double logarithmic stability estimate in the identification of a scalar potential by a partial elliptic Dirichlet-to-Neumann map, Bulletin of the South Ural State University, Ser. Mathematical Modelling, Programming and Computer Software (SUSU MMCS) 8(3) (2015), 78–95.Google Scholar
Choulli, M., Kian, Y. and Soccorsi, E., Stability result for elliptic inverse periodic coefficient problem by partial Dirichlet-to-Neumann map, J. Spectr. Theory 8(2) (2018), 733–768.Google Scholar
Choulli, M., Kian, Y. and Soccorsi, E., On the Calderón problem in periodic cylindrical domain with partial Dirichlet and Neumann data, Math. Methods Appl. Sci. 40 (2017), 5959–5974.Google Scholar
Choulli, M. and Soccorsi, E., An inverse anisotropic conductivity problem induced by twisting a homogeneous cylindrical domain, J. Spectr. Theory 5 (2015), 295–329.Google Scholar
Hörmander, L., The Analysis of Linear Partial Differential Operators, vol. II (Springer, Berlin, Heidelberg, 1983).Google Scholar
Hu, G. and Kian, Y., Determination of singular time-dependent coefficients for wave equations from full and partial data, Inverse Probl. Imaging 12 (2018), 745–772.Google Scholar
Ikehata, M., Inverse conductivity problem in the infinite slab, Inverse Problems 17 (2001), 437–454.Google Scholar
Imanuvilov, O., Uhlmann, G. and Yamamoto, M., The Calderón problem with partial data in two dimensions, J. Amer. Math. Soc. 23 (2010), 655–691.Google Scholar
Imanuvilov, O., Uhlmann, G. and Yamamoto, M., Partial Cauchy data for general second order elliptic operators in two dimensions, Publ. Res. Inst. Math. Sci. 48 (2012), 971–1055.Google Scholar
Isakov, V., Completness of products of solutions and some inverse problems for PDE, J. Differential Equations 92 (1991), 305–316.Google Scholar
Isakov, V., On uniqueness in the inverse conductivity problem with local data, Inverse Probl. Imaging 1 (2007), 95–105.Google Scholar
Kane, C., Balents, L. and Fisher, M. P. A., Coulomb interactions and mesoscopic effects in carbon nanotubes, Phys. Rev. Lett. 79 (1997), 5086–5089.Google Scholar
Kavian, O., Kian, Y. and Soccorsi, E., Uniqueness and stability results for an inverse spectral problem in a periodic waveguide, J. Math. Pures Appl. (9) 104(6) (2015), 1160–1189.Google Scholar
Kenig, C. E. and Salo, M., The Calderón problem with partial data on manifolds and applications, Anal. PDE 6(8) (2013), 2003–2048.Google Scholar
Kenig, C. E., Sjöstrand, J. and Uhlmann, G., The Calderon problem with partial data, Ann. of Math. (2) 165 (2007), 567–591.Google Scholar
Kian, Y., Stability of the determination of a coefficient for wave equations in an infinite waveguide, Inverse Probl. Imaging 8(3) (2014), 713–732.Google Scholar
Kian, Y., Unique determination of a time-dependent potential for wave equations from partial data, Ann. l’IHP (C) Nonlinear Anal. 34 (2017), 973–990.Google Scholar
Kian, Y., Recovery of time-dependent damping coefficients and potentials appearing in wave equations from partial data, SIAM J. Math. Anal. 48(6) (2016), 4021–4046.Google Scholar
Kian, Y. and Oksanen, L., Recovery of time-dependent coefficient on Riemannian manifold for hyperbolic equations, Int. Math. Res. Not. IMRN, to appear,https://doi.org/10.1093/imrn/rnx263.Google Scholar
Kian, Y., Phan, Q. S. and Soccorsi, E., Carleman estimate for infinite cylindrical quantum domains and application to inverse problems, Inverse Problems 30(5) (2014), 055016.Google Scholar
Kian, Y., Phan, Q. S. and Soccorsi, E., Hölder stable determination of a quantum scalar potential in unbounded cylindrical domains, J. Math. Anal. Appl. 426(1) (2015), 194–210.Google Scholar
Klibanov, M. V., Convexification of restricted Dirichlet-to-Neumann map, J. Inverse Ill-Posed Probl. 25 (2017), 669–685.Google Scholar
Krupchyk, K., Lassas, M. and Uhlmann, G., Inverse problems with partial data for a magnetic Schrödinger operator in an infinite slab or bounded domain, Comm. Math. Phys. 312 (2012), 87–126.Google Scholar
Li, X., Inverse boundary value problems with partial data in unbounded domains, Inverse Problems 28 (2012), 085003.Google Scholar
Li, X., Inverse problem for Schrödinger equations with Yang–Mills potentials in a slab, J. Differential Equations 253 (2012), 694–726.Google Scholar
Li, X. and Uhlmann, G., Inverse problems on a slab, Inverse Probl. Imaging 4 (2010), 449–462.Google Scholar
Lions, J.-L. and Magenes, E., Non-homogeneous Boundary Value Problems and Applications, vol. I (Dunod, Paris, 1968).Google Scholar
Nachman, A. and Street, B., Reconstruction in the Calderón problem with partial data, Commun. Partial Differential Equations 35 (2010), 375–390.Google Scholar
Potenciano-Machado, L., Stability estimates for a Magnetic Schrodinger operator with partial data. Preprint, 2016, arXiv:1610.04015.Google Scholar
Potenciano-Machado, L., Optimal stability estimates for a Magnetic Schrödinger operator with local data, Inverse Problems 33 (2017), 095001.Google Scholar
Salo, M. and Wang, J. N., Complex spherical waves and inverse problems in unbounded domains, Inverse Problems 22 (2006), 2299–2309.Google Scholar
Saut, J. C. and Scheurer, B., Sur l’unicité du problème de Cauchy et le prolongement unique pour des équations elliptiques à coefficients non localement bornés, J. Differential Equations 43 (1982), 28–43.Google Scholar
Sylvester, J. and Uhlmann, G., A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2) 125 (1987), 153–169.Google Scholar
Yang, Y., Determining the first order perturbation of a bi-harmonic operator on bounded and unbounded domains from partial data, J. Differential Equations 257 (2014), 3607–3639.Google Scholar