-
Previous Article
Extended backward stochastic Volterra integral equations and their applications to time-Inconsistent stochastic recursive control problems
- MCRF Home
- This Issue
-
Next Article
Local contact sub-Finslerian geometry for maximum norms in dimension 3
Stable determination of a vector field in a non-Self-Adjoint dynamical Schrödinger equation on Riemannian manifolds
1. | Université Tunis El Manar, Ecole Nationale d'ingénieurs de Tunis, ENIT-LAMSIN, B.P. 37, 1002 Tunis, Tunisia |
2. | Beijing Computational Science Research Center, Beijing 100193, China, and, Université Tunis El Manar, Faculté des Sciences de Tunis & ENIT-LAMSIN, B.P. 37, 1002 Tunis, Tunisia |
This paper deals with an inverse problem for a non-self-adjoint Schrödinger equation on a compact Riemannian manifold. Our goal is to stably determine a real vector field from the dynamical Dirichlet-to-Neumann map. We establish in dimension $ n\geq2 $, an Hölder type stability estimate for the inverse problem under study. The proof is mainly based on the reduction to an equivalent problem for an electro-magnetic Schrödinger equation and the use of a Carleman estimate designed for elliptic operators.
References:
[1] |
M. Bellassoued, Stable determination of coefficients in the dynamical Schrödinger equation in a magnetic field, Inverse Problems, 33 (2017), 055009, 36 pp.
doi: 10.1088/1361-6420/aa5fc5. |
[2] |
M. Bellassoued and I. Ben Aïcha, Optimal stability for a first order coefficient in a non-self-adjoint wave equation from Dirichlet-to-Neumann map, Inverse Problems, 33 (2017), 105006, 23 pp.
doi: 10.1088/1361-6420/aa8415. |
[3] |
M. Bellassoued and H. Benjoud,
Stability estimate for an inverse problem for the wave equation in a magnetic field, Appl. Anal, 87 (2008), 277-292.
doi: 10.1080/00036810801911264. |
[4] |
M. Bellassoued and M. Choulli,
Stability estimate for an inverse problem for the magnetic Schroödinger equation from the Dirichlet-to-Neumann map, J. Funct. Anal., 258 (2010), 161-195.
doi: 10.1016/j.jfa.2009.06.010. |
[5] |
M. Bellassoued and Z. Rezig,
Simultaneous determination of two coefficients in the Riemannian hyperbolic equation from boundary measurements, Ann. Global Anal. Geom., 56 (2019), 291-325.
doi: 10.1007/s10455-019-09668-7. |
[6] |
M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer, Tokyo, 2017.
doi: 10.1007/978-4-431-56600-7. |
[7] |
H. Ben Joud, A stability estimate for an inverse problem for the Schrödinger equation in a magnetic field from partial boundary measurements, Inverse Problems, 25 (2009), 045012, 23 pp.
doi: 10.1088/0266-5611/25/4/045012. |
[8] |
J. Cheng, G. Nakamura and E. Somersalo,
Uniqueness of identifying the convection term, Communications of the Korean Mathematical Society, 16 (2001), 405-413.
|
[9] |
N. S. Dairbekov,
Integral geometry problem for nontrapping manifolds, Inverse Problems, 22 (2006), 431-445.
doi: 10.1088/0266-5611/22/2/003. |
[10] |
Y. Kian and E. Soccorsi,
Hölder stably determining the time-dependent electromagnetic potential of the Schrödinger equation, SIAM J. Math. Anal., 51 (2019), 627-647.
doi: 10.1137/18M1197308. |
[11] |
Y. Kian and A. Tetlow,
Hölder stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic Schrödinger equation, Inverse Probl. Imaging, 14 (2020), 819-839.
doi: 10.3934/ipi.2020038. |
[12] |
K. Krupchyk and G. Uhlmann,
Inverse problems for advection diffusion equations in admissible geometries, Comm. Partial Differential Equations, 43 (2018), 585-615.
doi: 10.1080/03605302.2018.1446163. |
[13] |
Y. V. Kurylev and M. Lassas,
The multidimensional Gel'fand inverse problem for non-self-adjoint operators, Inverse Problems, 13 (1997), 1495-1501.
doi: 10.1088/0266-5611/13/6/006. |
[14] |
R. G. Muhometov,
The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry, Dokl. Akad. Nauk SSSR, 232 (1977), 32-35.
|
[15] |
V. Pohjola,
A uniqueness result for an inverse problem of the steady state convection diffusion equation, SIAM J. Math. Anal., 47 (2015), 2084-2103.
doi: 10.1137/140970926. |
[16] |
L. Pestov and G. Uhlmann, On characterization of the range and inversion of formulas for the geodesic $X$-ray transform, Int. Math. Res. Not., 2004 (2004) 4331–4347.
doi: 10.1155/S1073792804142116. |
[17] |
M. Salo, Inverse problems for nonsmooth first order perturbation of the Laplacian, Ann. Acad. Sci. Fenn. Math. Diss., 139 (2004), 67 pp. |
[18] |
V. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, 1994.
doi: 10.1515/9783110900095. |
[19] |
P. Stefanov and G. Uhlmann,
Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map, Int. Math. Res. Not., 17 (2005), 1047-1061.
doi: 10.1155/IMRN.2005.1047. |
[20] |
P. Stefanov and G. Uhlmann,
Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J., 123 (2004), 445-467.
doi: 10.1215/S0012-7094-04-12332-2. |
[21] |
Z. Q. Sun,
An inverse boundary value problem for Schrödinger operators with vector potentials, Trans. Amer. Math. Soc., 388 (1993), 953-969.
doi: 10.2307/2154438. |
show all references
References:
[1] |
M. Bellassoued, Stable determination of coefficients in the dynamical Schrödinger equation in a magnetic field, Inverse Problems, 33 (2017), 055009, 36 pp.
doi: 10.1088/1361-6420/aa5fc5. |
[2] |
M. Bellassoued and I. Ben Aïcha, Optimal stability for a first order coefficient in a non-self-adjoint wave equation from Dirichlet-to-Neumann map, Inverse Problems, 33 (2017), 105006, 23 pp.
doi: 10.1088/1361-6420/aa8415. |
[3] |
M. Bellassoued and H. Benjoud,
Stability estimate for an inverse problem for the wave equation in a magnetic field, Appl. Anal, 87 (2008), 277-292.
doi: 10.1080/00036810801911264. |
[4] |
M. Bellassoued and M. Choulli,
Stability estimate for an inverse problem for the magnetic Schroödinger equation from the Dirichlet-to-Neumann map, J. Funct. Anal., 258 (2010), 161-195.
doi: 10.1016/j.jfa.2009.06.010. |
[5] |
M. Bellassoued and Z. Rezig,
Simultaneous determination of two coefficients in the Riemannian hyperbolic equation from boundary measurements, Ann. Global Anal. Geom., 56 (2019), 291-325.
doi: 10.1007/s10455-019-09668-7. |
[6] |
M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer, Tokyo, 2017.
doi: 10.1007/978-4-431-56600-7. |
[7] |
H. Ben Joud, A stability estimate for an inverse problem for the Schrödinger equation in a magnetic field from partial boundary measurements, Inverse Problems, 25 (2009), 045012, 23 pp.
doi: 10.1088/0266-5611/25/4/045012. |
[8] |
J. Cheng, G. Nakamura and E. Somersalo,
Uniqueness of identifying the convection term, Communications of the Korean Mathematical Society, 16 (2001), 405-413.
|
[9] |
N. S. Dairbekov,
Integral geometry problem for nontrapping manifolds, Inverse Problems, 22 (2006), 431-445.
doi: 10.1088/0266-5611/22/2/003. |
[10] |
Y. Kian and E. Soccorsi,
Hölder stably determining the time-dependent electromagnetic potential of the Schrödinger equation, SIAM J. Math. Anal., 51 (2019), 627-647.
doi: 10.1137/18M1197308. |
[11] |
Y. Kian and A. Tetlow,
Hölder stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic Schrödinger equation, Inverse Probl. Imaging, 14 (2020), 819-839.
doi: 10.3934/ipi.2020038. |
[12] |
K. Krupchyk and G. Uhlmann,
Inverse problems for advection diffusion equations in admissible geometries, Comm. Partial Differential Equations, 43 (2018), 585-615.
doi: 10.1080/03605302.2018.1446163. |
[13] |
Y. V. Kurylev and M. Lassas,
The multidimensional Gel'fand inverse problem for non-self-adjoint operators, Inverse Problems, 13 (1997), 1495-1501.
doi: 10.1088/0266-5611/13/6/006. |
[14] |
R. G. Muhometov,
The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry, Dokl. Akad. Nauk SSSR, 232 (1977), 32-35.
|
[15] |
V. Pohjola,
A uniqueness result for an inverse problem of the steady state convection diffusion equation, SIAM J. Math. Anal., 47 (2015), 2084-2103.
doi: 10.1137/140970926. |
[16] |
L. Pestov and G. Uhlmann, On characterization of the range and inversion of formulas for the geodesic $X$-ray transform, Int. Math. Res. Not., 2004 (2004) 4331–4347.
doi: 10.1155/S1073792804142116. |
[17] |
M. Salo, Inverse problems for nonsmooth first order perturbation of the Laplacian, Ann. Acad. Sci. Fenn. Math. Diss., 139 (2004), 67 pp. |
[18] |
V. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, 1994.
doi: 10.1515/9783110900095. |
[19] |
P. Stefanov and G. Uhlmann,
Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map, Int. Math. Res. Not., 17 (2005), 1047-1061.
doi: 10.1155/IMRN.2005.1047. |
[20] |
P. Stefanov and G. Uhlmann,
Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J., 123 (2004), 445-467.
doi: 10.1215/S0012-7094-04-12332-2. |
[21] |
Z. Q. Sun,
An inverse boundary value problem for Schrödinger operators with vector potentials, Trans. Amer. Math. Soc., 388 (1993), 953-969.
doi: 10.2307/2154438. |
[1] |
Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094 |
[2] |
Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021 |
[3] |
Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271 |
[4] |
Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261 |
[5] |
M. Phani Sudheer, Ravi S. Nanjundiah, A. S. Vasudeva Murthy. Revisiting the slow manifold of the Lorenz-Krishnamurthy quintet. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1403-1416. doi: 10.3934/dcdsb.2006.6.1403 |
[6] |
Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209 |
[7] |
Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017 |
[8] |
Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166 |
[9] |
Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 |
[10] |
Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022 |
[11] |
Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212 |
[12] |
Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089 |
[13] |
Guangying Lv, Jinlong Wei, Guang-an Zou. Noise and stability in reaction-diffusion equations. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021005 |
[14] |
Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463 |
[15] |
Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367 |
[16] |
Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450 |
[17] |
Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021066 |
[18] |
Liangliang Ma. Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021068 |
[19] |
Akio Matsumot, Ferenc Szidarovszky. Stability switching and its directions in cournot duopoly game with three delays. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021069 |
[20] |
Michel Chipot, Mingmin Zhang. On some model problem for the propagation of interacting species in a special environment. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020401 |
2019 Impact Factor: 0.857
Tools
Article outline
[Back to Top]