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

[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

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

[1]

Mourad Bellassoued, David Dos Santos Ferreira. Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map. Inverse Problems & Imaging, 2011, 5 (4) : 745-773. doi: 10.3934/ipi.2011.5.745

[2]

Victor Isakov, Jenn-Nan Wang. Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map. Inverse Problems & Imaging, 2014, 8 (4) : 1139-1150. doi: 10.3934/ipi.2014.8.1139

[3]

Jussi Behrndt, A. F. M. ter Elst. The Dirichlet-to-Neumann map for Schrödinger operators with complex potentials. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 661-671. doi: 10.3934/dcdss.2017033

[4]

Victor Isakov. Increasing stability for the Schrödinger potential from the Dirichlet-to Neumann map. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 631-640. doi: 10.3934/dcdss.2011.4.631

[5]

Mahamadi Warma. A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2043-2067. doi: 10.3934/cpaa.2015.14.2043

[6]

Wolfgang Arendt, Rafe Mazzeo. Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2201-2212. doi: 10.3934/cpaa.2012.11.2201

[7]

Kevin Arfi, Anna Rozanova-Pierrat. Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 1-26. doi: 10.3934/dcdss.2019001

[8]

Ihsane Bikri, Ronald B. Guenther, Enrique A. Thomann. The Dirichlet to Neumann map - An application to the Stokes problem in half space. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 221-230. doi: 10.3934/dcdss.2010.3.221

[9]

Francis J. Chung. Partial data for the Neumann-Dirichlet magnetic Schrödinger inverse problem. Inverse Problems & Imaging, 2014, 8 (4) : 959-989. doi: 10.3934/ipi.2014.8.959

[10]

Robert Carlson. Dirichlet to Neumann maps for infinite quantum graphs. Networks & Heterogeneous Media, 2012, 7 (3) : 483-501. doi: 10.3934/nhm.2012.7.483

[11]

Shouchuan Hu, Nikolaos S. Papageorgiou. Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2889-2922. doi: 10.3934/cpaa.2013.12.2889

[12]

Monique Dauge, Thomas Ourmières-Bonafos, Nicolas Raymond. Spectral asymptotics of the Dirichlet Laplacian in a conical layer. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1239-1258. doi: 10.3934/cpaa.2015.14.1239

[13]

Carmen Calvo-Jurado, Juan Casado-Díaz, Manuel Luna-Laynez. Parabolic problems with varying operators and Dirichlet and Neumann boundary conditions on varying sets. Conference Publications, 2007, 2007 (Special) : 181-190. doi: 10.3934/proc.2007.2007.181

[14]

Sébastien Gouëzel. An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1205-1208. doi: 10.3934/dcds.2009.24.1205

[15]

Plamen Stefanov, Gunther Uhlmann, Andras Vasy. On the stable recovery of a metric from the hyperbolic DN map with incomplete data. Inverse Problems & Imaging, 2016, 10 (4) : 1141-1147. doi: 10.3934/ipi.2016035

[16]

Wolfgang Arendt, Daniel Daners. Varying domains: Stability of the Dirichlet and the Poisson problem. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 21-39. doi: 10.3934/dcds.2008.21.21

[17]

Haitao Yang, Yibin Zhang. Boundary bubbling solutions for a planar elliptic problem with exponential Neumann data. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5467-5502. doi: 10.3934/dcds.2017238

[18]

Keonhee Lee, Ngoc-Thach Nguyen, Yinong Yang. Topological stability and spectral decomposition for homeomorphisms on noncompact spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2487-2503. doi: 10.3934/dcds.2018103

[19]

Matthew O. Williams, Clarence W. Rowley, Ioannis G. Kevrekidis. A kernel-based method for data-driven koopman spectral analysis. Journal of Computational Dynamics, 2015, 2 (2) : 247-265. doi: 10.3934/jcd.2015005

[20]

Massimiliano Guzzo, Giancarlo Benettin. A spectral formulation of the Nekhoroshev theorem and its relevance for numerical and experimental data analysis. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 1-28. doi: 10.3934/dcdsb.2001.1.1

2018 Impact Factor: 1.469

Metrics

  • PDF downloads (66)
  • HTML views (111)
  • Cited by (0)

Other articles
by authors

[Back to Top]