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Stable Determination of an Inclusion in a Layered Medium with Special Anisotropy

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Mathematical and Numerical Approaches for Multi-Wave Inverse Problems (CIRM 2019)

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Abstract

In this note we review some recent results concerning the inverse inclusion problem. In particular we analyze the stability issue for defect contained in layered medium where the conductivity is different in each layer. We consider conductivities with special anisotropy. The modulus of continuity obtained is of logarithmic type, which as shown in Di Cristo and Rondi (Inverse Prob 19:685–701 [13]) turns out to be optimal.

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References

  1. Alessandrini, G.: Singular solutions of elliptic equations and the determination of conductivity by boundary measurements. J. Differ. Eqn. 84, 252–272 (1990)

    Google Scholar 

  2. Alessandrini, G., Di Cristo, M.: Stable determanation of an inclusion by boundary measurements SIAM. J. Math. Anal. 37, 200–217 (2005)

    MATH  Google Scholar 

  3. Alessandrini, G., Di Cristo, M., Morassi, A., Rosset, E.: Stable determination of an inclusion in an elastic body by boundary measurements SIAM. J. Math. Anal. 46, 2692–2729 (2014)

    MathSciNet  MATH  Google Scholar 

  4. Alessandrini, G., Sincich, E.: Cracks with impedance, stable determination from boundary data Indiana Univ. Math. J. 62(3), 947–989 (2013)

    MathSciNet  MATH  Google Scholar 

  5. Calderon, A.: On an inverse boundary value problem. Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), pp. 65–73, Soc. Brasil. Mat., Rio de Janeiro (1980)

    Google Scholar 

  6. Carstea, A., Wang, J.N.: Propagation of smallness for an elliptic PDE with piecewise Lipschitz ceofficients. J. Differ. Eqn 268, 7609-7628 (2020)

    Google Scholar 

  7. Di Cristo, M.: Stable determination of an inhomogeneous inclusion by local boundary measurements. J. Comput. Appl. Math. 198, 414–425 (2007)

    Article  MathSciNet  Google Scholar 

  8. Di Cristo, M.: Stability estimates in the inverse transmission scattering problem. Inverse Probl. Imaging 3, 551–565 (2009)

    Article  MathSciNet  Google Scholar 

  9. Di Cristo, M., Francini, E, Lin, C.L., Vessella, S., Wang, J.N.: Carleman estimate for second order elliptic equations with Lipschitz leading coefficients and jumps at an interface. J. Math. Pures Appl. 108, 163–206 (2017)

    Google Scholar 

  10. Di Cristo, M., Ren, Y.: Stable determination of an inclusion for a class of anisotropic conductivities. Inverse Probl. 33, 15 (2017)

    Google Scholar 

  11. Di Cristo, M., Ren, Y.: Stable determination of an inclusion in a layered medium. Math. Methods Appl. Sci. 41, 4602–4611 (2018)

    Article  MathSciNet  Google Scholar 

  12. Di Cristo, M., Ren, Y.: Three sphere inequality for second order elliptic equations with coefficients with jump discontinuity. J. Diff. Eqn. 266, 936–941 (2019)

    Article  MathSciNet  Google Scholar 

  13. Di Cristo, M., Rondi, L.: Examples of exponential instability for inverse inclusion and scattering problems. Inverse Prob. 19, 685–701 (2003)

    Article  MathSciNet  Google Scholar 

  14. Francini, E., Lin, C.-L., Vessella, S., Wang, J.-N.: Three-region inequalities for the second order elliptic equation with discontinuous coefficients and size estimate. J. Diff. Eqn. 261, 5306–5323 (2016)

    Article  MathSciNet  Google Scholar 

  15. Isakov, V.: On uniqueness of recovery of a discontinuous conductivity coefficient. Comm. Pure Appl. Math. 41, 865–877 (1988)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Politecnico di Milano, Milan, Italy

    Michele Di Cristo

Authors
  1. Michele Di Cristo
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Corresponding author

Correspondence to Michele Di Cristo .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden

    Larisa Beilina

  2. Départment de Mathématiques - MAPMO, University of Orléans, Orleans, France

    Maïtine Bergounioux

  3. Institut de Mathématiques de Marseille, Aix-Marseille University, Marseille, France

    Michel Cristofol

  4. Institut Fresnel, Marseille, France

    Anabela Da Silva

  5. Institut Fresnel, Marseille, France

    Amelie Litman

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Di Cristo, M. (2020). Stable Determination of an Inclusion in a Layered Medium with Special Anisotropy. In: Beilina, L., Bergounioux, M., Cristofol, M., Da Silva, A., Litman, A. (eds) Mathematical and Numerical Approaches for Multi-Wave Inverse Problems. CIRM 2019. Springer Proceedings in Mathematics & Statistics, vol 328. Springer, Cham. https://doi.org/10.1007/978-3-030-48634-1_3

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