May  2009, 3(2): 319-332. doi: 10.3934/ipi.2009.3.319

Recovering an obstacle using integral equations


Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, United States

Received  January 2009 Revised  March 2009 Published  May 2009

We consider the inverse problem of recovering the shape, location and surface properties of an object where the surrounding medium is both conductive and homogeneous and we measure Cauchy data on an accessible part of the exterior boundary. It is assumed that the physical situation is modelled by harmonic functions and the boundary condition on the obstacle is one of Dirichlet type. The purpose of this paper is to answer some of the questions raised in a recent paper that introduced a nonlinear integral equation approach for the solution of this type of problem.
Citation: William Rundell. Recovering an obstacle using integral equations. Inverse Problems and Imaging, 2009, 3 (2) : 319-332. doi: 10.3934/ipi.2009.3.319

Shitao Liu, Roberto Triggiani. Determining damping and potential coefficients of an inverse problem for a system of two coupled hyperbolic equations. Part I: Global uniqueness. Conference Publications, 2011, 2011 (Special) : 1001-1014. doi: 10.3934/proc.2011.2011.1001


Jun Lai, Ming Li, Peijun Li, Wei Li. A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers. Inverse Problems and Imaging, 2018, 12 (3) : 635-665. doi: 10.3934/ipi.2018027


Benjamin Boutin, Frédéric Coquel, Philippe G. LeFloch. Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure. Networks and Heterogeneous Media, 2021, 16 (2) : 283-315. doi: 10.3934/nhm.2021007


Brian Sleeman. The inverse acoustic obstacle scattering problem and its interior dual. Inverse Problems and Imaging, 2009, 3 (2) : 211-229. doi: 10.3934/ipi.2009.3.211


Laurent Bourgeois, Dmitry Ponomarev, Jérémi Dardé. An inverse obstacle problem for the wave equation in a finite time domain. Inverse Problems and Imaging, 2019, 13 (2) : 377-400. doi: 10.3934/ipi.2019019


Laurent Bourgeois, Jérémi Dardé. A quasi-reversibility approach to solve the inverse obstacle problem. Inverse Problems and Imaging, 2010, 4 (3) : 351-377. doi: 10.3934/ipi.2010.4.351


Laurent Bourgeois, Jérémi Dardé. The "exterior approach" to solve the inverse obstacle problem for the Stokes system. Inverse Problems and Imaging, 2014, 8 (1) : 23-51. doi: 10.3934/ipi.2014.8.23


Lekbir Afraites. A new coupled complex boundary method (CCBM) for an inverse obstacle problem. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 23-40. doi: 10.3934/dcdss.2021069


Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380


Fioralba Cakoni, Rainer Kress. Integral equations for inverse problems in corrosion detection from partial Cauchy data. Inverse Problems and Imaging, 2007, 1 (2) : 229-245. doi: 10.3934/ipi.2007.1.229


Fabio Camilli, Paola Loreti, Naoki Yamada. Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1291-1302. doi: 10.3934/cpaa.2009.8.1291


J.I. Díaz, D. Gómez-Castro. Steiner symmetrization for concave semilinear elliptic and parabolic equations and the obstacle problem. Conference Publications, 2015, 2015 (special) : 379-386. doi: 10.3934/proc.2015.0379


Weisong Dong, Tingting Wang, Gejun Bao. A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1769-1780. doi: 10.3934/cpaa.2016013


M. R. Arias, R. Benítez. Properties of solutions for nonlinear Volterra integral equations. Conference Publications, 2003, 2003 (Special) : 42-47. doi: 10.3934/proc.2003.2003.42


Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems and Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121


Peijun Li, Xiaokai Yuan. Inverse obstacle scattering for elastic waves in three dimensions. Inverse Problems and Imaging, 2019, 13 (3) : 545-573. doi: 10.3934/ipi.2019026


Masaru Ikehata, Esa Niemi, Samuli Siltanen. Inverse obstacle scattering with limited-aperture data. Inverse Problems and Imaging, 2012, 6 (1) : 77-94. doi: 10.3934/ipi.2012.6.77


Lu Zhao, Heping Dong, Fuming Ma. Inverse obstacle scattering for acoustic waves in the time domain. Inverse Problems and Imaging, 2021, 15 (5) : 1269-1286. doi: 10.3934/ipi.2021037


Frank Hettlich. The domain derivative for semilinear elliptic inverse obstacle problems. Inverse Problems and Imaging, 2022, 16 (4) : 691-702. doi: 10.3934/ipi.2021071


Hongxia Zhang, Ying Wang. Liouville results for fully nonlinear integral elliptic equations in exterior domains. Communications on Pure and Applied Analysis, 2018, 17 (1) : 85-112. doi: 10.3934/cpaa.2018006

2021 Impact Factor: 1.483


  • PDF downloads (72)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]