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Elliptic Equations: Single Boundary Measurements

  • Victor Isakov
Chapter
Part of the Applied Mathematical Sciences book series (AMS, volume 127)

Abstract

In this chapter we consider the elliptic second-order differential equation \(Au = f\quad \mathrm{in}\;\varOmega,f = f_{0} -\sum \limits _{j=1}^{n}\partial _{j}f_{j}\) with the Dirichlet boundary data \(u = g_{0}\quad \mathrm{on}\;\partial \varOmega.\) We assume that A = div(−a∇) + b ⋅ ∇ + c with bounded and measurable coefficients a (symmetric real-valued (n × n) matrix) and complex-valued b and c in L (Ω). Another assumption is that A is an elliptic operator; i.e., there is ɛ0 > 0 such that a(x)ξ ⋅ ξ ≥ ɛ0 | ξ | 2 for any vector \(\xi \in \mathbb{R}^{n}\) and any x ∈ Ω. Unless specified otherwise, we assume that Ω is a bounded domain in \(\mathbb{R}^{n}\) with the boundary of class C2. However, most of the results are valid for Lipschitz boundaries.

References

  1. [ADN]
    Agmon, S., Douglis, A., Nirenberg, L. Estimates near the boundary for the solutions of elliptic differential equations satisfying general boundary values, I. Comm. Pure Appl. Math., 12 (1959), 623–727.MathSciNetCrossRefMATHGoogle Scholar
  2. [Ah]
    Ahlfors, L. Complex Analysis. McGraw-Hill, 1979.MATHGoogle Scholar
  3. [Al2]
    Alessandrini, G. Remark on a paper of Bellout and Friedman. Boll. Unione Mat. Ital., (7) 3A (1989), 243–250.Google Scholar
  4. [Al4]
    Alessandrini, G. Examples of instability in inverse boundary value problems. Inverse Problems, 13 (1997), 887–897.MathSciNetCrossRefMATHGoogle Scholar
  5. [AlBRV]
    Alessandrini, G., Beretta, E., Rosset, E., Vessella, S. Optimal stability for Inverse Elliptic Boundary Value Problems. Ann. Sc. Norm. Sup. Pisa, 29 (2000), 755–806.MathSciNetMATHGoogle Scholar
  6. [AlD]
    Alessandrini, G., DiBenedetto, E. Determining 2-dimensional cracks in 3-dimensional bodies: uniqueness and stability. Indiana Univ. Math. J., 46 (1997), 1–83.MathSciNetMATHGoogle Scholar
  7. [AlIP]
    Alessandrini, G., Isakov, V., Powell, J. Local Uniqueness in the Inverse Conductivity Problem with One Measurement. Trans. of AMS, 347 (1995), 3031–3041.MathSciNetCrossRefMATHGoogle Scholar
  8. [AlM]
    Alessandrini, G., Magnanini, R. The Index of Isolated Critical Points and Solutions of Elliptic Equations in the Plane. Ann. Sc., Norm. Pisa IV, 19 (1992), 567–591.Google Scholar
  9. [AlRS]
    Alessandrini, G., Rosset, E., Seo, J.K. Optimal size estimates for the inverse conductivity with one measurement. Proc. AMS, 128 (1999), 53–64.MathSciNetCrossRefMATHGoogle Scholar
  10. [AlV]
    Alessandrini, G., Valenzuela, A.D. Unique determination of multiple cracks by two measurements. SIAM J. Control Opt., 34 (1996), 913–921.MathSciNetCrossRefMATHGoogle Scholar
  11. [AK]
    Ammari, H., Kang, H. Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory. Springer-Verlag, 2007.MATHGoogle Scholar
  12. [AnB]
    Andrieux, S., Ben Abda, A. Identification de fissures planes par une donn’ee de bord unique: une procédé direct de localisation et d’identification. C.R. Acad. Sc. Paris, t.315. ser. 1 (1992), 1323–1328.Google Scholar
  13. [B]
    Bacchelli, V. Uniqueness for determination of unknown boundary and impedance with the homogeneous Robin condition. Inverse Problems, 25 (2009), 015004.MathSciNetCrossRefMATHGoogle Scholar
  14. [BerFV]
    Beretta, E., Francini, E., Vogelius, M. Asymptotic formulas for steady state voltage potentials in the presence of thin inhomogeneities. A rigorous error analysis. J. Math. Pures Appl., 82 (2003), 1277–1301.Google Scholar
  15. [BerVo]
    Beretta, E., Vogelius, M. An Inverse Problem originating from magnetohy-drodynamics. II, Domain with arbitrary corners. Asymptotic Anal., 11 (1995), 289–315.Google Scholar
  16. [BrV]
    Bryan, K., Vogelius, M. Effective behavior of clusters of microscopic cracks inside a homogeneous conductor. Asympt. Anal., 16 (1998), 141–179.MathSciNetMATHGoogle Scholar
  17. [BuK]
    Bukhgeim, A.L., Klibanov, M.V. Uniqueness in the large of a class of multidimensional inverse problems. Soviet Math. Dokl. 24 (1981), 244–247.Google Scholar
  18. [BuEMP]
    Burger, M., Engl, H.W., Markowich, P.A., Pietra, P. Identification of doping profiles in semiconductor devices. Inverse Problems, 17 (2001), 1765–1796.MathSciNetCrossRefMATHGoogle Scholar
  19. [Cher]
    Cherednichenko, V. Inverse logarithmic potential problem. VSP, 1996.MATHGoogle Scholar
  20. [DM]
    Demidov, A.S., Moussaoui, M. An inverse problem originating from magnetohydrodynamics. Inverse Problems, 20 (2004), 137–154.MathSciNetCrossRefMATHGoogle Scholar
  21. [DR]
    Di Cristo, M., Rondi, L. Examples of exponential instability for inverse inclusion and scattering problems. Inverse Problems, 19 (2003), 685–701.MathSciNetCrossRefMATHGoogle Scholar
  22. [ElcIN]
    Elcrat, A., Isakov, V., Neculoiu, O. On finding a surface crack from boundary measurements. Inverse Problems, 11 (1995), 343–351.MathSciNetCrossRefMATHGoogle Scholar
  23. [ElcIKS]
    Elcrat, A., Isakov, V., Kropf, E., Stewart, D. A stability analysis of the harmonic continuation. Inverse Problems, 28 (2012), 075016.MathSciNetCrossRefMATHGoogle Scholar
  24. [Ell]
    Eller, M. Identification of cracks in three-dimensional bodies by many boundary measurements. Inverse Problems, 12 (1996), 395–408.MathSciNetCrossRefMATHGoogle Scholar
  25. [EsFV]
    Escauriaza, L., Fabes, E., Verchota, G. On a regularity theorem for weak solutions to transmission problems with internal Lipschitz boundaries. Proc. AMS., 115 (1992), 1069–1076.MathSciNetCrossRefMATHGoogle Scholar
  26. [FI]
    Friedman, A., Isakov, V. On the Uniqueness in the Inverse Conductivity Problem with one Measurement. Indiana Univ. Math. J., 38 (1989), 553–580.MathSciNetMATHGoogle Scholar
  27. [FV]
    Friedman, A., Vogelius, M. Determining Cracks by Boundary Measurements. Indiana Univ. Math. J., 38 (1989), 497–525.MathSciNetCrossRefMATHGoogle Scholar
  28. [I3]
    Inverse Problems: Theory and Applications Alessandrini, G., Uhlmann, G., editors. Contemp. Math., 333, AMS, Providence, RI, 2003.Google Scholar
  29. [Is4]
    Isakov, V. Inverse Source Problems. Math. Surveys and Monographs Series, Vol. 34, AMS, Providence, R.I., 1990.Google Scholar
  30. [IsLQ]
    Isakov, V., Leung, S., Qian, J. A three-dimensional inverse gravimetry problem for ice with snow caps. Inv. Probl. Imag., 7 (2013), 523–545.MathSciNetCrossRefMATHGoogle Scholar
  31. [IsM]
    Isakov, V., Myers, J. On the inverse doping profile problem. Inv. Probl. Imag, 6 (2012), 465–486.MathSciNetCrossRefMATHGoogle Scholar
  32. [IsP]
    Isakov, V., Powell, J. On inverse conductivity problem with one measurement. Inverse Problems, 6 (1990), 311–318.MathSciNetCrossRefMATHGoogle Scholar
  33. [Iv]
    Ivanov, V.K. An integral equation of the inverse problem of the logarithmic potential. Dokl. Akad. Nauk SSSR, 105 (1955), 409–411.MathSciNetGoogle Scholar
  34. [J4]
    John, F. Partial Differential Equations. Springer, 1982.CrossRefMATHGoogle Scholar
  35. [KS1]
    Kang, H., Seo, J.K. The layer potential technique for the inverse conductivity problem. Inverse Problems, 12 (1996), 267–278.MathSciNetCrossRefMATHGoogle Scholar
  36. [KS2]
    Kang, H., Seo, J.K. Inverse conductivity problem with one measurement: uniqueness of balls in \(\mathbb{R}^{3}\). SIAM J. Appl. Math., 59 (1999), 1533–1539.MathSciNetCrossRefMATHGoogle Scholar
  37. [Kh1]
    Khaidarov, A. A class of inverse problems for elliptic equations. Diff. Equat., 23 (1987), 939–945.MathSciNetMATHGoogle Scholar
  38. [KiS]
    Kim, H., Seo, J.K. Unique determination of a collection of a finite number of cracks from two boundary measurements. SIAM J. Math. Anal., 27 (1996), 1336–1340.MathSciNetCrossRefMATHGoogle Scholar
  39. [KinS]
    Kinderlehrer, D., Stampacchia, G. An introduction to variational inequalities and their applications. Academic Press, 1980.MATHGoogle Scholar
  40. [KlT]
    Klibanov, M.V., Timonov, A. Carleman Estimates for Coefficient Inverse Problems and Numerical Applications. VSP, Utrecht, The Netherlands, 2004.CrossRefMATHGoogle Scholar
  41. [LU]
    Ladyzhenskaya, O.A., Ural’tseva, N.N. Linear and Quasilinear Elliptic Equations. Academic Press, New York-London, 1969.MATHGoogle Scholar
  42. [Mi]
    Miranda, C. Partial Differential Equations of Elliptic Type. Ergebn. Math., Band 2, Springer-Verlag, 1970.Google Scholar
  43. [Mor]
    Morrey, C.B., Jr. Multiple Integrals in the Calculus of Variations. Springer, 1966.MATHGoogle Scholar
  44. [Mus]
    Muskhelisvili, N.I. Singular Integral Equations. Noordhoff, Groningen, 1953.Google Scholar
  45. [No]
    Novikov, P.S. Sur le problème inverse du potentiel. Dokl. Akad. Nauk SSSR, 18 (1938), 165–168.MATHGoogle Scholar
  46. [Pow]
    Powell, J. On a small perturbation in the two dimensional inverse conductivity problem. J. Math. Anal. Appl., 175 (1993), 292–304.MathSciNetCrossRefMATHGoogle Scholar
  47. [Pr]
    Prilepko, A.I. Über die Existenz and Eindeutigkeit von Lösungen inverser Probleme. Math Nach., 63 (1974), 135–153.CrossRefMATHGoogle Scholar
  48. [PrOV]
    Prilepko, A.I., Orlovskii, D.G., Vasin, I.A. Methods for solving inverse problems in mathematical physics. Marcel Dekker, New York-Basel, 2000.Google Scholar
  49. [Ro]
    Robbiano, L. Theorème d’Unicité Adapté au Contrôle des Solutions des Problèmes Hyperboliques. Comm. Part. Diff. Equat., 16 (1991), 789–801.CrossRefGoogle Scholar
  50. [Sa]
    Sakai, M. Regularity of a boundary having a Schwarz function. Acta Mathematica, 166 (1991), 263–287.MathSciNetCrossRefMATHGoogle Scholar
  51. [Se]
    Seo, J.K. On uniqueness in the inverse conductivity problem. J. Fourier Anal. Appl., 2 (1996), 227–235.MathSciNetCrossRefMATHGoogle Scholar
  52. [SoZ]
    Sokolowski, J., Zolesio, J.-P. Introduction to Shape Optimization. Berlin, Springer-Verlag, 1992.CrossRefMATHGoogle Scholar
  53. [V]
    Vekua, I. N. Generalized analytic functions. Pergamon Press, London, 1962.MATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Victor Isakov
    • 1
  1. 1.Department of Mathematics and StatisticsWichita State UniversityWichitaUSA

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