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August  2020, 14(4): 631-664. doi: 10.3934/ipi.2020029

Uniqueness results in the inverse spectral Steklov problem

Germain Gendron ,

Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629, 2 Rue de la Houssinière BP 92208, F-44322 Nantes Cedex 03, France

Received  June 2019 Revised  January 2020 Published  May 2020

This paper is devoted to an inverse Steklov problem for a particular class of $ n $-dimensional manifolds having the topology of a hollow sphere and equipped with a warped product metric. We prove that the knowledge of the Steklov spectrum determines uniquely the associated warping function up to a natural invariance.

Keywords: Inverse Calderón problem, Steklov spectrum, Weyl-Titchmarsh functions, Nevanlinna theorem, local Borg-Marchenko theorem.
Mathematics Subject Classification: Primary: 58J53, 35P20; Secondary: 30D35.
Citation: Germain Gendron. Uniqueness results in the inverse spectral Steklov problem. Inverse Problems & Imaging, 2020, 14 (4) : 631-664. doi: 10.3934/ipi.2020029
References:
[1]

M. S. Agranovich, On a mixed Poincaré-Steklov type spectral problem in a Lipschitz domain, Russ. J. Math. Phys., 13 (2006), 239-244.  doi: 10.1134/S1061920806030010.  Google Scholar

[2]

B. Colbois, A. Girouard and A. Hassannezhad, The Steklov and Laplacian spectra of Riemannian manifolds with boundary, J. Funct. Anal., 278 (2020), 108409. doi: 10.1016/j.jfa.2019.108409.  Google Scholar

[3]

T. Daudé, N. Kamran and F. Nicoleau, Non-uniqueness results in the anisotropic Calderón problem with data measured on disjoint sets, Annales de l'Institut Fourier, 69 (2015). Google Scholar

[4]

T. Daudé, N. Kamran and F. Nicoleau, On the hidden mechanism behind non-uniqueness for the anisotropic Calderón problem with data on disjoint sets, Ann. Henri Poincaré, 20 (2019), 859-887.  doi: 10.1007/s00023-018-00755-2.  Google Scholar

[5]

T. Daudé, N. Kamran and F. Nicoleau, Stability in the inverse Steklov problem on warped product Riemannian manifolds, preprint, arXiv: 1812.07235. Google Scholar

[6]

D. D. S. Ferreira, C. Kenig, M. Salo and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Invent. Math., 178 (2009), 119-171.  doi: 10.1007/s00222-009-0196-4.  Google Scholar

[7]

A. Girouard, J. Lagacé, I. Polterovich and A. Savo, The Steklov spectrum of cuboids, Mathematika, 65 (2019), 272-310.  doi: 10.1112/S0025579318000414.  Google Scholar

[8]

A. Girouard and I. Polterovich, Spectral geometry of the Steklov problem (survey article), J. Spectr. Theory, 7 (2017), 321-360.  doi: 10.4171/JST/164.  Google Scholar

[9]

A. Jollivet and V. Sharafutdinov, On an inverse problem for the Steklov spectrum of a Riemannian surface, Contemp. Math., 615 (2014), 165-191.  doi: 10.1090/conm/615/12260.  Google Scholar

[10]

R. V. Kohn and M. Vogelius, Identification of an unknown conductivity by means of measurements at the boundary, SIAM-AMS Proc., 14 (1984), 113-123.   Google Scholar

[11]

W. R. B. Lionheart, Conformal uniqueness results in anisotropic electrical impedance imaging, Inverse Problems, 13 (1997), 125-134.  doi: 10.1088/0266-5611/13/1/010.  Google Scholar

[12]

O. Parzanchevski, On G-sets and isospectrality, Ann. Inst. Fourier, 63 (2013), 2307-2329.  doi: 10.5802/aif.2831.  Google Scholar

[13]

J. Pöschel and E. Trubowitz, Inverse Spectral Theory, in Pure and Applied Mathematics, 130, Academic Press, Inc., Boston, MA, 1987.  Google Scholar

[14]

L. Provenzano and J. Stubbe, Weyl-type bounds for Steklov eigenvalues, J. Spectr. Theory, 9 (2019) 349–377. doi: 10.4171/JST/250.  Google Scholar

[15]

A. G. Ramm, An inverse scattering problem with part of the fixed-energy phase shifts, Comm. Math. Phys., 207 (1999), 231-247.  doi: 10.1007/s002200050725.  Google Scholar

[16]

Y. Safarov and D. Vassilev, The Asymptotic Distribution of Eigenvalues of Partial Differential Operators, American Mathematical Society, Providence, RI, 1997.  Google Scholar

[17]

M. Salo, The Calderón problem on Riemannian manifolds, in Inverse Problems and Applications: Inside Out. II, Math. Sci. Res. Inst. Publ., 60, Cambridge Univ. Press, Cambridge, 2013,167–247.  Google Scholar

[18]

M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-96854-9.  Google Scholar

[19]

B. Simon, A new approach to inverse spectral theory. Ⅰ. Fundamental formalism, Ann. of Math., 150 (1999), 1029-1057.  doi: 10.2307/121061.  Google Scholar

[20]

G. Uhlmann, Recent progress in the anisotropic electrical impedance problem, in Proceedings of the USA-Chile Workshop on Nonlinear Analysis (Viña del Mar-Valparaiso, 2000), J. Diff. Eqns., Conf., 6, Southwest Texas State Univ., San Marcos, TX, 2001,303–311.  Google Scholar

[21]

G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse problems, 25 (2009) 123011, 39 pp. doi: 10.1088/0266-5611/25/12/123011.  Google Scholar

show all references

References:
[1]

M. S. Agranovich, On a mixed Poincaré-Steklov type spectral problem in a Lipschitz domain, Russ. J. Math. Phys., 13 (2006), 239-244.  doi: 10.1134/S1061920806030010.  Google Scholar

[2]

B. Colbois, A. Girouard and A. Hassannezhad, The Steklov and Laplacian spectra of Riemannian manifolds with boundary, J. Funct. Anal., 278 (2020), 108409. doi: 10.1016/j.jfa.2019.108409.  Google Scholar

[3]

T. Daudé, N. Kamran and F. Nicoleau, Non-uniqueness results in the anisotropic Calderón problem with data measured on disjoint sets, Annales de l'Institut Fourier, 69 (2015). Google Scholar

[4]

T. Daudé, N. Kamran and F. Nicoleau, On the hidden mechanism behind non-uniqueness for the anisotropic Calderón problem with data on disjoint sets, Ann. Henri Poincaré, 20 (2019), 859-887.  doi: 10.1007/s00023-018-00755-2.  Google Scholar

[5]

T. Daudé, N. Kamran and F. Nicoleau, Stability in the inverse Steklov problem on warped product Riemannian manifolds, preprint, arXiv: 1812.07235. Google Scholar

[6]

D. D. S. Ferreira, C. Kenig, M. Salo and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Invent. Math., 178 (2009), 119-171.  doi: 10.1007/s00222-009-0196-4.  Google Scholar

[7]

A. Girouard, J. Lagacé, I. Polterovich and A. Savo, The Steklov spectrum of cuboids, Mathematika, 65 (2019), 272-310.  doi: 10.1112/S0025579318000414.  Google Scholar

[8]

A. Girouard and I. Polterovich, Spectral geometry of the Steklov problem (survey article), J. Spectr. Theory, 7 (2017), 321-360.  doi: 10.4171/JST/164.  Google Scholar

[9]

A. Jollivet and V. Sharafutdinov, On an inverse problem for the Steklov spectrum of a Riemannian surface, Contemp. Math., 615 (2014), 165-191.  doi: 10.1090/conm/615/12260.  Google Scholar

[10]

R. V. Kohn and M. Vogelius, Identification of an unknown conductivity by means of measurements at the boundary, SIAM-AMS Proc., 14 (1984), 113-123.   Google Scholar

[11]

W. R. B. Lionheart, Conformal uniqueness results in anisotropic electrical impedance imaging, Inverse Problems, 13 (1997), 125-134.  doi: 10.1088/0266-5611/13/1/010.  Google Scholar

[12]

O. Parzanchevski, On G-sets and isospectrality, Ann. Inst. Fourier, 63 (2013), 2307-2329.  doi: 10.5802/aif.2831.  Google Scholar

[13]

J. Pöschel and E. Trubowitz, Inverse Spectral Theory, in Pure and Applied Mathematics, 130, Academic Press, Inc., Boston, MA, 1987.  Google Scholar

[14]

L. Provenzano and J. Stubbe, Weyl-type bounds for Steklov eigenvalues, J. Spectr. Theory, 9 (2019) 349–377. doi: 10.4171/JST/250.  Google Scholar

[15]

A. G. Ramm, An inverse scattering problem with part of the fixed-energy phase shifts, Comm. Math. Phys., 207 (1999), 231-247.  doi: 10.1007/s002200050725.  Google Scholar

[16]

Y. Safarov and D. Vassilev, The Asymptotic Distribution of Eigenvalues of Partial Differential Operators, American Mathematical Society, Providence, RI, 1997.  Google Scholar

[17]

M. Salo, The Calderón problem on Riemannian manifolds, in Inverse Problems and Applications: Inside Out. II, Math. Sci. Res. Inst. Publ., 60, Cambridge Univ. Press, Cambridge, 2013,167–247.  Google Scholar

[18]

M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-96854-9.  Google Scholar

[19]

B. Simon, A new approach to inverse spectral theory. Ⅰ. Fundamental formalism, Ann. of Math., 150 (1999), 1029-1057.  doi: 10.2307/121061.  Google Scholar

[20]

G. Uhlmann, Recent progress in the anisotropic electrical impedance problem, in Proceedings of the USA-Chile Workshop on Nonlinear Analysis (Viña del Mar-Valparaiso, 2000), J. Diff. Eqns., Conf., 6, Southwest Texas State Univ., San Marcos, TX, 2001,303–311.  Google Scholar

[21]

G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse problems, 25 (2009) 123011, 39 pp. doi: 10.1088/0266-5611/25/12/123011.  Google Scholar

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