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: |
[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. Bellassoued, M. 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. Bellassoued, M. 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. Blåsten, O. 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. Imanuvilov, J. 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. Imanuvilov, G. 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. Kurylev, M. 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. Nachman, J. 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.
![]() ![]() |