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Stability estimates for time-dependent coefficients appearing in the magnetic Schrödinger equation from arbitrary boundary measurements
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Numerical recovery of magnetic diffusivity in a three dimensional spherical dynamo equation
Hölder-stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic Schrödinger equation
1. | Aix-Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France |
2. | Department of Mathematics, University College London, London, UK |
We consider the inverse problem of Hölder-stably determining the time- and space-dependent coefficients of the Schrödinger equation on a simple Riemannian manifold with boundary of dimension $ n\geq2 $ from the knowledge of the Dirichlet-to-Neumann map. Assuming the divergence of the magnetic potential is known, we show that the electric and magnetic potentials can be Hölder-stably recovered from these data. Here we also remove the smallness assumption for the solenoidal part of the magnetic potential present in previous results.
References:
[1] |
Y. E. Anikonov and V. G. Romanov,
On uniqueness of determination of a form of first degree by its integrals along geodesics, J. Inv. Ill-Posed Problems, 5 (1997), 487-480.
doi: 10.1515/jiip.1997.5.6.487. |
[2] |
M. Bellassoued, Stable determination of coefficients in the dynamical Schrödinger equation in a magnetic field, Inverse Problems, 33 (2017), 36pp.
doi: 10.1088/1361-6420/aa5fc5. |
[3] |
M. Bellassoued and M. Choulli,
Stability estimate for an inverse problem for the magnetic Schrödinger equation from the Dirichlet-to-Neumann map, J. Funct. Anal., 258 (2010), 161-195.
doi: 10.1016/j.jfa.2009.06.010. |
[4] |
M. Bellassoued and D. D. S. Ferreira, Stable determination of coefficients in the dynamical anisotropic Schrödinger equation from the Dirichlet-to-Neumann map, Inverse Problems, 26 (2010), 125010.
doi: 10.1088/0266-5611/26/12/125010. |
[5] |
M. Bellassoued, Y. Kian and E. Soccorsi,
An inverse problem for the magnetic Schrödinger equation in infinite cylindrical domains, Publications of the Research Institute for Mathematical Sciences, 54 (2018), 679-728.
doi: 10.4171/PRIMS/54-4-1. |
[6] |
M. Bellassoued and Z. Rezig,
Simultaneous determination of two coefficients in the Riemannian hyperbolic equation from boundary measurements, Ann. Glob. Anal. Geom., 56 (2019), 291-325.
doi: 10.1007/s10455-019-09668-7. |
[7] |
I. B. Aicha, Stability estimate for an inverse problem for the Schrödinger equation in a magnetic field with time-dependent coefficient, Journal of Mathematical Physics, 58 (2017), 071508.
doi: 10.1063/1.4995606. |
[8] |
M. Choulli, Y. Kian and E. Soccorsi,
Stable determination of time-dependent scalar potential from boundary measurements in a periodic quantum waveguide, SIAM J. Math. Anal., 47 (2015), 4536-4558.
doi: 10.1137/140986268. |
[9] |
G. Eskin, Inverse problems for the Schrödinger equations with time-dependent electromagnetic potentials and the Aharonov-Bohm effect, J. Math. Phys., 49 (2008), 022105.
doi: 10.1063/1.2841329. |
[10] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985. |
[11] |
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. |
[12] |
J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, Springer, 1997.
doi: 10.1007/b98852. |
[13] |
J-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Dunod, Paris, 1968. |
[14] |
C. Montalto,
Stable determination of a simple metric, a covector field and a potential from the hyperbolic Dirichlet-to-Neumann map, Communications in Partial Differential Equations, 39 (2012), 120-145.
doi: 10.1080/03605302.2013.843429. |
[15] |
M. Salo, Inverse problems for nonsmooth first order perturbations of the Laplacian, Ann. Acad. Scient. Fenn. Math. Dissertations, 139 (2004). |
[16] |
V. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, the Netherlands, 1994.
doi: 10.1515/9783110900095. |
[17] |
M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish Inc., USA, 1970. |
[18] |
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. |
[19] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.
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show all references
References:
[1] |
Y. E. Anikonov and V. G. Romanov,
On uniqueness of determination of a form of first degree by its integrals along geodesics, J. Inv. Ill-Posed Problems, 5 (1997), 487-480.
doi: 10.1515/jiip.1997.5.6.487. |
[2] |
M. Bellassoued, Stable determination of coefficients in the dynamical Schrödinger equation in a magnetic field, Inverse Problems, 33 (2017), 36pp.
doi: 10.1088/1361-6420/aa5fc5. |
[3] |
M. Bellassoued and M. Choulli,
Stability estimate for an inverse problem for the magnetic Schrödinger equation from the Dirichlet-to-Neumann map, J. Funct. Anal., 258 (2010), 161-195.
doi: 10.1016/j.jfa.2009.06.010. |
[4] |
M. Bellassoued and D. D. S. Ferreira, Stable determination of coefficients in the dynamical anisotropic Schrödinger equation from the Dirichlet-to-Neumann map, Inverse Problems, 26 (2010), 125010.
doi: 10.1088/0266-5611/26/12/125010. |
[5] |
M. Bellassoued, Y. Kian and E. Soccorsi,
An inverse problem for the magnetic Schrödinger equation in infinite cylindrical domains, Publications of the Research Institute for Mathematical Sciences, 54 (2018), 679-728.
doi: 10.4171/PRIMS/54-4-1. |
[6] |
M. Bellassoued and Z. Rezig,
Simultaneous determination of two coefficients in the Riemannian hyperbolic equation from boundary measurements, Ann. Glob. Anal. Geom., 56 (2019), 291-325.
doi: 10.1007/s10455-019-09668-7. |
[7] |
I. B. Aicha, Stability estimate for an inverse problem for the Schrödinger equation in a magnetic field with time-dependent coefficient, Journal of Mathematical Physics, 58 (2017), 071508.
doi: 10.1063/1.4995606. |
[8] |
M. Choulli, Y. Kian and E. Soccorsi,
Stable determination of time-dependent scalar potential from boundary measurements in a periodic quantum waveguide, SIAM J. Math. Anal., 47 (2015), 4536-4558.
doi: 10.1137/140986268. |
[9] |
G. Eskin, Inverse problems for the Schrödinger equations with time-dependent electromagnetic potentials and the Aharonov-Bohm effect, J. Math. Phys., 49 (2008), 022105.
doi: 10.1063/1.2841329. |
[10] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985. |
[11] |
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. |
[12] |
J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, Springer, 1997.
doi: 10.1007/b98852. |
[13] |
J-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Dunod, Paris, 1968. |
[14] |
C. Montalto,
Stable determination of a simple metric, a covector field and a potential from the hyperbolic Dirichlet-to-Neumann map, Communications in Partial Differential Equations, 39 (2012), 120-145.
doi: 10.1080/03605302.2013.843429. |
[15] |
M. Salo, Inverse problems for nonsmooth first order perturbations of the Laplacian, Ann. Acad. Scient. Fenn. Math. Dissertations, 139 (2004). |
[16] |
V. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, the Netherlands, 1994.
doi: 10.1515/9783110900095. |
[17] |
M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish Inc., USA, 1970. |
[18] |
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. |
[19] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.
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