Search Advanced

Mathematics in Engineering

2021, Volume 3, Issue 1: 1-10. doi: 10.3934/mine.2021003
Previous Article Next Article
Research article Special Issues

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.


    加载中


    [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.
  • Reader Comments
  • © 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)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematics in Engineering
1.4 2.2

Metrics

Article views(3628) PDF downloads(616) Cited by(0)

Preview PDF
Download XML
Export Citation
Article outline

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog