December  2019, 13(6): 1213-1258. doi: 10.3934/ipi.2019054

Stability for determination of Riemannian metrics by spectral data and Dirichlet-to-Neumann map limited on arbitrary subboundary

1. 

Department of Mathematics, Colorado State University, 101 Weber Building, Fort Collins, CO 80523-1874, USA

2. 

Department of Mathematical Sciences, The University of Tokyo, Komaba, Meguro, Tokyo 153, Japan

3. 

Honorary Member of Academy of Romanian Scientists, Splaiul Independentei Street, no 54, 050094 Bucharest Romania

4. 

Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

* Corresponding author: Masairo Yamamoto

dedicated to the memory of Professor Yaroslav Kurylev

Received  September 2018 Revised  April 2019 Published  October 2019

Fund Project: The first author is supported by NSF grant DMS 13129000.
The second author is supported by Grant-in-Aid for Scientific Research (S) 15H05740 of Japan Society for the Promotion of Science and by The National Natural Science Foundation of China (no. 11771270, 91730303), and prepared with the support of the "RUDN University Program 5-100".

In this paper, we establish conditional stability estimates for two inverse problems of determining metrics in two dimensional Laplace-Beltrami operators. As data, in the first inverse problem we adopt spectral data on an arbitrarily fixed subboundary, while in the second, we choose the Dirichlet-to-Neumann map limited on an arbitrarily fixed subboundary. The conditional stability estimates for the two inverse problems are stated as follows. If the difference between spectral data or Dirichlet-to-Neumann maps related to two metrics $ {\bf{g}}_1 $ and $ {\bf{g}}_2 $ is small, then $ {\bf{g}}_1 $ and $ {\bf{g}}_2 $ are close in $ L^2(\Omega) $ modulo a suitable diffeomorphism within a priori bounds of $ {\bf{g}}_1 $ and $ {\bf{g}}_2 $. Both stability estimates are of the same double logarithmic rate.

Citation: Oleg Yu. Imanuvilov, Masahiro Yamamoto. Stability for determination of Riemannian metrics by spectral data and Dirichlet-to-Neumann map limited on arbitrary subboundary. Inverse Problems & Imaging, 2019, 13 (6) : 1213-1258. doi: 10.3934/ipi.2019054
References:
[1]

G. Alessandrini, L. Rondi, E. Rosset and S. Vessela, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 123004. Google Scholar

[2]

K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton Mathematical Series, 48. Princeton University Press, Princeton, NJ, 2009.  Google Scholar

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M. I. Belishev and Y. V. Kurylev, To the reconstruction of a Riemanninan manifold via its spectral data (BC-method), Comm. Partial Differential Equations, 17 (1992), 767-804.  doi: 10.1080/03605309208820863.  Google Scholar

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M. BellassouedM. Choulli and M. Yamamoto, Stability estimate for an inverse wave equation and a multidimensional Borg-Levinson theorem, J. Differential Equations, 247 (2009), 465-494.  doi: 10.1016/j.jde.2009.03.024.  Google Scholar

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M. BellassouedM. Choulli and M. Yamamoto, Stability estimate for a multidimensional inverse spectral problem with partial spectral data, J. Math. Anal. Appl., 378 (2011), 184-197.  doi: 10.1016/j.jmaa.2011.01.007.  Google Scholar

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M. Bellasoued and D. Dos Santos Ferreira, Stability estimates for the anisotropic wave equations from the Dirichlet-to-Neumann map, Inverse Probl. Imaging, 5 (2011), 745-773.  doi: 10.3934/ipi.2011.5.745.  Google Scholar

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E. BlåstenO. Y. Imanuvilov and M. Yamamoto, Stability and uniqueness for a two-dimensional inverse boundary value problem for less regular potentials, Inverse Probl. Imaging, 9 (2015), 709-723.  doi: 10.3934/ipi.2015.9.709.  Google Scholar

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M. Choulli and P. Stefanov, Stability for the multi-dimensional Borg-Levinson theorem with partial spectral data, Comm. Partial Differential Equations, 38 (2013), 455-476.  doi: 10.1080/03605302.2012.747538.  Google Scholar

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L. C. Evans, Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1998. doi: 10.1090/gsm/019.  Google Scholar

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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

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R. E. Greene and S. G. Krantz, Function Theory of One Complex Variable, Second edition, Graduate Studies in Mathematics, 40. Amer. Math. Soc., Providence, RI, 2002.  Google Scholar

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O. Y. Imanuvilov, Controllability of parabolic equations, Math. Sb., 186 (1995), 879-900.  doi: 10.1070/SM1995v186n06ABEH000047.  Google Scholar

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O. Y. ImanuvilovJ. P. Puel and M. Yamamoto, Carleman estimate for parabolic equations with nonhomogeneous boundary conditions, Chin. Ann. Math., 30 (2009), 333-378.  doi: 10.1007/s11401-008-0280-x.  Google Scholar

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O. Y. ImanuvilovG. Uhlmann and M. Yamamoto, Partial Cauchy data for general second order operators in two dimensions, Publ. Res. Inst. Math. Sci., 48 (2012), 971-1055.  doi: 10.2977/PRIMS/94.  Google Scholar

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O. Y. Imanuvilov and M. Yamamoto, Uniqueness for inverse boundary value problems by Dirichlet-to-Neumann map on subboundaries, Milan J. Math., 81 (2013), 187-258.  doi: 10.1007/s00032-013-0205-3.  Google Scholar

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H. Isozaki, Some remarks on the multi-dimensional Borg-Levinson theorem, J. Math. Kyoto Univ., 31 (1991), 743-753.  doi: 10.1215/kjm/1250519727.  Google Scholar

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A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123. Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420036220.  Google Scholar

[19]

Y. KurylevM. Lassas and R. Weder, Multidimensional Borg-Levinson theorem, Inverse Problems, 21 (2005), 1685-1696.  doi: 10.1088/0266-5611/21/5/011.  Google Scholar

[20]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, Berlin-New York-Heidelberg, 1972.  Google Scholar

[21]

N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435-1444.  doi: 10.1088/0266-5611/17/5/313.  Google Scholar

[22]

C. Montalto, Stable determination of a simple metric, a covector field and a potential from the hyperbolic Dirichlet-to-Neumann map, Comm. Partial Differential Equations, 39 (2014), 120-145.  doi: 10.1080/03605302.2013.843429.  Google Scholar

[23]

A. NachmanJ. Sylvester and G. Uhlmann, An $n$-dimensional Borg-Levinson theorem, Comm. Math. Phys., 115 (1988), 595-605.  doi: 10.1007/BF01224129.  Google Scholar

[24]

R. G. Novikov, A multidimensional inverse spectral problem for the equation $-\Delta \psi + (v(x)-Eu(x))\psi = 0$, Functional Analysis and its Applications, 22 (1988), 263-272.  doi: 10.1007/BF01077418.  Google Scholar

[25]

R. G. Novikov and M. Santacesaria, A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions, J. Inverse Ill-Posed Probl., 18 (2010), 765-785.  doi: 10.1515/JIIP.2011.003.  Google Scholar

[26]

R. G. Novikov and M. Santacesaria, Global uniqueness and reconstruction for the multi-channel Gel'fand-Calderón inverse problem in two dimensions, Bull. Sci. Math., 135 (2011), 421-434.  doi: 10.1016/j.bulsci.2011.04.007.  Google Scholar

[27]

L. Päivärinta and V. Serov, An $n$-dimensional Borg-Levinson theorem for singular potentials, Adv. in Appl. Math., 29 (2002), 509-520.  doi: 10.1016/S0196-8858(02)00027-1.  Google Scholar

[28]

C. Pommeranke, Boundary Behavior of Conformal Mappings, Grundlehren der Mathematischen Wissenschaften, Band 299. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-02770-7.  Google Scholar

[29]

M. Santacesaria, New global stability estimates for the Calderón problem in two dimensions, J. Inst. Math. Jussieu, 12 (2013), 553-569.  doi: 10.1017/S147474801200076X.  Google Scholar

[30]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993.  Google Scholar

[31]

J. Sylvester, An anisotropic inverse boundary value problem, Communications on Pure and Applied Mathematics, 43 (1990), 201-232.  doi: 10.1002/cpa.3160430203.  Google Scholar

[32]

I. N. Vekua, Generalized Analytic Functions, Pergamon Press, London-Paris-Frankfurt, Addison-Wesley Publishing Co., Inc., Reading, Mass. 1962.  Google Scholar

show all references

References:
[1]

G. Alessandrini, L. Rondi, E. Rosset and S. Vessela, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 123004. Google Scholar

[2]

K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton Mathematical Series, 48. Princeton University Press, Princeton, NJ, 2009.  Google Scholar

[3]

M. I. Belishev, Boundary control in reconstruction of manifolds and metrics (the BC method), Inverse Problems, 13 (1997), R1–R45. doi: 10.1088/0266-5611/13/5/002.  Google Scholar

[4]

M. I. Belishev and Y. V. Kurylev, To the reconstruction of a Riemanninan manifold via its spectral data (BC-method), Comm. Partial Differential Equations, 17 (1992), 767-804.  doi: 10.1080/03605309208820863.  Google Scholar

[5]

M. BellassouedM. Choulli and M. Yamamoto, Stability estimate for an inverse wave equation and a multidimensional Borg-Levinson theorem, J. Differential Equations, 247 (2009), 465-494.  doi: 10.1016/j.jde.2009.03.024.  Google Scholar

[6]

M. BellassouedM. Choulli and M. Yamamoto, Stability estimate for a multidimensional inverse spectral problem with partial spectral data, J. Math. Anal. Appl., 378 (2011), 184-197.  doi: 10.1016/j.jmaa.2011.01.007.  Google Scholar

[7]

M. Bellasoued and D. Dos Santos Ferreira, Stability estimates for the anisotropic wave equations from the Dirichlet-to-Neumann map, Inverse Probl. Imaging, 5 (2011), 745-773.  doi: 10.3934/ipi.2011.5.745.  Google Scholar

[8]

E. BlåstenO. Y. Imanuvilov and M. Yamamoto, Stability and uniqueness for a two-dimensional inverse boundary value problem for less regular potentials, Inverse Probl. Imaging, 9 (2015), 709-723.  doi: 10.3934/ipi.2015.9.709.  Google Scholar

[9]

M. Choulli and P. Stefanov, Stability for the multi-dimensional Borg-Levinson theorem with partial spectral data, Comm. Partial Differential Equations, 38 (2013), 455-476.  doi: 10.1080/03605302.2012.747538.  Google Scholar

[10]

L. C. Evans, Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1998. doi: 10.1090/gsm/019.  Google Scholar

[11]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[12]

R. E. Greene and S. G. Krantz, Function Theory of One Complex Variable, Second edition, Graduate Studies in Mathematics, 40. Amer. Math. Soc., Providence, RI, 2002.  Google Scholar

[13]

O. Y. Imanuvilov, Controllability of parabolic equations, Math. Sb., 186 (1995), 879-900.  doi: 10.1070/SM1995v186n06ABEH000047.  Google Scholar

[14]

O. Y. ImanuvilovJ. P. Puel and M. Yamamoto, Carleman estimate for parabolic equations with nonhomogeneous boundary conditions, Chin. Ann. Math., 30 (2009), 333-378.  doi: 10.1007/s11401-008-0280-x.  Google Scholar

[15]

O. Y. ImanuvilovG. Uhlmann and M. Yamamoto, Partial Cauchy data for general second order operators in two dimensions, Publ. Res. Inst. Math. Sci., 48 (2012), 971-1055.  doi: 10.2977/PRIMS/94.  Google Scholar

[16]

O. Y. Imanuvilov and M. Yamamoto, Uniqueness for inverse boundary value problems by Dirichlet-to-Neumann map on subboundaries, Milan J. Math., 81 (2013), 187-258.  doi: 10.1007/s00032-013-0205-3.  Google Scholar

[17]

H. Isozaki, Some remarks on the multi-dimensional Borg-Levinson theorem, J. Math. Kyoto Univ., 31 (1991), 743-753.  doi: 10.1215/kjm/1250519727.  Google Scholar

[18]

A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123. Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420036220.  Google Scholar

[19]

Y. KurylevM. Lassas and R. Weder, Multidimensional Borg-Levinson theorem, Inverse Problems, 21 (2005), 1685-1696.  doi: 10.1088/0266-5611/21/5/011.  Google Scholar

[20]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, Berlin-New York-Heidelberg, 1972.  Google Scholar

[21]

N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435-1444.  doi: 10.1088/0266-5611/17/5/313.  Google Scholar

[22]

C. Montalto, Stable determination of a simple metric, a covector field and a potential from the hyperbolic Dirichlet-to-Neumann map, Comm. Partial Differential Equations, 39 (2014), 120-145.  doi: 10.1080/03605302.2013.843429.  Google Scholar

[23]

A. NachmanJ. Sylvester and G. Uhlmann, An $n$-dimensional Borg-Levinson theorem, Comm. Math. Phys., 115 (1988), 595-605.  doi: 10.1007/BF01224129.  Google Scholar

[24]

R. G. Novikov, A multidimensional inverse spectral problem for the equation $-\Delta \psi + (v(x)-Eu(x))\psi = 0$, Functional Analysis and its Applications, 22 (1988), 263-272.  doi: 10.1007/BF01077418.  Google Scholar

[25]

R. G. Novikov and M. Santacesaria, A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions, J. Inverse Ill-Posed Probl., 18 (2010), 765-785.  doi: 10.1515/JIIP.2011.003.  Google Scholar

[26]

R. G. Novikov and M. Santacesaria, Global uniqueness and reconstruction for the multi-channel Gel'fand-Calderón inverse problem in two dimensions, Bull. Sci. Math., 135 (2011), 421-434.  doi: 10.1016/j.bulsci.2011.04.007.  Google Scholar

[27]

L. Päivärinta and V. Serov, An $n$-dimensional Borg-Levinson theorem for singular potentials, Adv. in Appl. Math., 29 (2002), 509-520.  doi: 10.1016/S0196-8858(02)00027-1.  Google Scholar

[28]

C. Pommeranke, Boundary Behavior of Conformal Mappings, Grundlehren der Mathematischen Wissenschaften, Band 299. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-02770-7.  Google Scholar

[29]

M. Santacesaria, New global stability estimates for the Calderón problem in two dimensions, J. Inst. Math. Jussieu, 12 (2013), 553-569.  doi: 10.1017/S147474801200076X.  Google Scholar

[30]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993.  Google Scholar

[31]

J. Sylvester, An anisotropic inverse boundary value problem, Communications on Pure and Applied Mathematics, 43 (1990), 201-232.  doi: 10.1002/cpa.3160430203.  Google Scholar

[32]

I. N. Vekua, Generalized Analytic Functions, Pergamon Press, London-Paris-Frankfurt, Addison-Wesley Publishing Co., Inc., Reading, Mass. 1962.  Google Scholar

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