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Mathematics in Engineering

2021, Volume 3, Issue 1: 1-10. doi: 10.3934/mine.2021003
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Lipschitz stable determination of small conductivity inclusions in a semilinear equation from boundary data

  • 1.
    Science Division, Experimental Research Building (C1), NYU Abu Dhabi, Saadiyat Campus, NYU Abu Dhabi, United Arab Emirates
  • 2.
    Department of Mathematics, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy
  • 3.
    Department of Mathematics and Statistics, University of Helsinki, Pietari Kalmin katu 5, 00560 Helsinki, Finland
  • Received: 02 March 2020 Accepted: 08 July 2020 Published: 23 July 2020

  • We consider an inverse problem regarding the detection of small conductivity inhomogeneities in a boundary value problem for a semilinear elliptic equation. For such a problem, that is related to cardiac electrophysiology, an asymptotic expansion for the boundary potential due to the presence of small conductivity inhomogeneities was established in [4]. Starting from this we derive Lipschitz continuous dependence estimates for the corresponding inverse problem.

    Citation: Elena Beretta, M. Cristina Cerutti, Luca Ratti. Lipschitz stable determination of small conductivity inclusions in a semilinear equation from boundary data[J]. Mathematics in Engineering, 2021, 3(1): 1-10. doi: 10.3934/mine.2021003

    Related Papers:

  • We consider an inverse problem regarding the detection of small conductivity inhomogeneities in a boundary value problem for a semilinear elliptic equation. For such a problem, that is related to cardiac electrophysiology, an asymptotic expansion for the boundary potential due to the presence of small conductivity inhomogeneities was established in [4]. Starting from this we derive Lipschitz continuous dependence estimates for the corresponding inverse problem.


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    [1] Alessandrini G, Rondi L, Rosset E, et al. (2009) The stability for the Cauchy problem for elliptic equations. Inverse Probl 25: 123004.
    [2] Ammari H, Kang H (2004) Reconstruction of Small Inhomogeneities from Boundary Measurements, Berlin: Springer.
    [3] Bacchelli V, Vessella S (2006) Lipschitz stability for a stationary 2D inverse problem with unknown polygonal boundary. Inverse Probl 22: 035013.
    [4] Beretta E, Cerutti MC, Manzoni A, et al. (2016) An asymptotic formula for boundary potential perturbations in a semilinear elliptic equation related to cardiac electrophysiology. Math Mod Meth Appl Sci 26: 645-670.
    [5] Beretta E, Manzoni A, Ratti L (2017) A reconstruction algorithm based on topological gradient for an inverse problem related to a semilinear elliptic boundary value problem. Inverse Probl 33: 035010.
    [6] Bourgeois B (2013) A remark on Lipschitz stability for inverse problems. C R Math 351: 187-190.
    [7] Capdeboscq Y, Vogelius M (2003) A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction. Math Model Num Anal 37: 159-173.
    [8] Cedio-Fengya D, Moskow S, Vogelius M (1998) Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction. Inverse Probl 14: 553-595.
    [9] Friedman A, Vogelius M (1989) Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction. Arch Rat Mech Anal 105: 299-326.
    [10] Miranda C (2013) Partial Differential Equations of Elliptic Type, Springer-Verlag.
    [11] Alessandrini G, Magnanini R (1994) Elliptic equations in divergence form, geometric critical points of solutions and Stekloff eigenfunctions. SIAM J Math Anal 25: 1259-1268.
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  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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