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Uniqueness results in the inverse spectral Steklov problem
Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629, 2 Rue de la Houssinière BP 92208, F-44322 Nantes Cedex 03, France |
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.
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. |
| [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. |
| [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). |
| [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. |
| [5] |
T. Daudé, N. Kamran and F. Nicoleau, Stability in the inverse Steklov problem on warped product Riemannian manifolds, preprint, arXiv: 1812.07235. |
| [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. |
| [7] |
A. Girouard, J. Lagacé, I. Polterovich and A. Savo,
The Steklov spectrum of cuboids, Mathematika, 65 (2019), 272-310.
doi: 10.1112/S0025579318000414. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [12] |
O. Parzanchevski,
On G-sets and isospectrality, Ann. Inst. Fourier, 63 (2013), 2307-2329.
doi: 10.5802/aif.2831. |
| [13] |
J. Pöschel and E. Trubowitz, Inverse Spectral Theory, in Pure and Applied Mathematics, 130, Academic Press, Inc., Boston, MA, 1987. |
| [14] |
L. Provenzano and J. Stubbe, Weyl-type bounds for Steklov eigenvalues, J. Spectr. Theory, 9 (2019) 349–377.
doi: 10.4171/JST/250. |
| [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. |
| [16] |
Y. Safarov and D. Vassilev, The Asymptotic Distribution of Eigenvalues of Partial Differential Operators, American Mathematical Society, Providence, RI, 1997. |
| [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. |
| [18] |
M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, 1987.
doi: 10.1007/978-3-642-96854-9. |
| [19] |
B. Simon,
A new approach to inverse spectral theory. Ⅰ. Fundamental formalism, Ann. of Math., 150 (1999), 1029-1057.
doi: 10.2307/121061. |
| [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. |
| [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. |
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. |
| [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. |
| [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). |
| [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. |
| [5] |
T. Daudé, N. Kamran and F. Nicoleau, Stability in the inverse Steklov problem on warped product Riemannian manifolds, preprint, arXiv: 1812.07235. |
| [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. |
| [7] |
A. Girouard, J. Lagacé, I. Polterovich and A. Savo,
The Steklov spectrum of cuboids, Mathematika, 65 (2019), 272-310.
doi: 10.1112/S0025579318000414. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [12] |
O. Parzanchevski,
On G-sets and isospectrality, Ann. Inst. Fourier, 63 (2013), 2307-2329.
doi: 10.5802/aif.2831. |
| [13] |
J. Pöschel and E. Trubowitz, Inverse Spectral Theory, in Pure and Applied Mathematics, 130, Academic Press, Inc., Boston, MA, 1987. |
| [14] |
L. Provenzano and J. Stubbe, Weyl-type bounds for Steklov eigenvalues, J. Spectr. Theory, 9 (2019) 349–377.
doi: 10.4171/JST/250. |
| [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. |
| [16] |
Y. Safarov and D. Vassilev, The Asymptotic Distribution of Eigenvalues of Partial Differential Operators, American Mathematical Society, Providence, RI, 1997. |
| [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. |
| [18] |
M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, 1987.
doi: 10.1007/978-3-642-96854-9. |
| [19] |
B. Simon,
A new approach to inverse spectral theory. Ⅰ. Fundamental formalism, Ann. of Math., 150 (1999), 1029-1057.
doi: 10.2307/121061. |
| [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. |
| [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. |
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