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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[31]

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

[32]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[31]

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

[32]

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

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