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On a logarithmic stability estimate for an inverse heat conduction problem
June
2019, 9(2): 289-312.
doi: 10.3934/mcrf.2019015
Application of the boundary control method to partial data Borg-Levinson inverse spectral problem
| 1. | Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France |
| 2. | Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France |
| 3. | Department of Mathematics, University College London, London, UK |
We consider the multidimensional Borg-Levinson problem of determining a potential $q$, appearing in the Dirichlet realization of the Schrödinger operator $A_q = -\Delta+q$ on a bounded domain $\Omega\subset\mathbb{R}^n$, $n\geq2$, from the boundary spectral data of $A_q$ on an arbitrary portion of $\partial\Omega$. More precisely, for $\gamma$ an open and non-empty subset of $\partial\Omega$, we consider the boundary spectral data on $\gamma$ given by ${\rm BSD}(q, \gamma): = \{(\lambda_{k}, {\partial_\nu \varphi_{k}}_{|\gamma}):\ k \geq1\}$, where $\{ \lambda_k:\ k \geq1\}$ is the non-decreasing sequence of eigenvalues of $A_q$, $\{ \varphi_k:\ k \geq1 \}$ an associated orthonormal basis of eigenfunctions, and $\nu$ is the unit outward normal vector to $\partial\Omega$. Our main result consists of determining a bounded potential $q\in L^\infty(\Omega)$ from the data ${\rm BSD}(q, \gamma)$. Previous uniqueness results, with arbitrarily small $\gamma$, assume that $q$ is smooth. Our approach is based on the Boundary Control method, and we give a self-contained presentation of the method, focusing on the analytic rather than geometric aspects of the method.
Keywords:
Inverse problems,
inverse spectral problem,
wave equation,
Boundary Control method,
uniqueness,
partial data,
unique continuation.
Mathematics Subject Classification:
35R30, 35J10, 35L05.
Citation:
Yavar Kian, Morgan Morancey, Lauri Oksanen. Application of the boundary control method to partial data Borg-Levinson inverse spectral problem. Mathematical Control and Related Fields,
2019, 9
(2)
: 289-312.
doi: 10.3934/mcrf.2019015
![]()
References:
| [1] |
C. Bardos, G. Lebeau and J. Rauch,
Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM Journal on Control and Optimization, 30 (1992), 1024-1065.
doi: 10.1137/0330055. |
| [2] |
M. Belishev,
An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297 (1987), 524-527.
|
| [3] |
M. Belishev and Y. Kurylev,
To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. Partial Differential Equations, 17 (1992), 767-804.
doi: 10.1080/03605309208820863. |
| [4] |
G. Borg,
Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Acta Math., 78 (1946), 1-96.
doi: 10.1007/BF02421600. |
| [5] |
B. Canuto and O. Kavian, Determining two coefficients in elliptic operators via boundary spectral data: A uniqueness result, Bolletino Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 7 (2004), 207–230. |
| [6] |
M. Choulli and P. Stefanov,
Stability for the multi-dimensional Borg-Levinson theorem with partial spectral data, Commun. Partial Diff. Eqns., 38 (2013), 455-476.
|
| [7] |
I. M. Gel'fand and B. M. Levitan,
On the determination of a differential equation from its spectral function, Izv. Akad. Nauk USSR, Ser. Mat., 15 (1951), 309-360.
|
| [8] |
L. Hörmander, Linear Partial Differential Operators, Springer Verlag, Berlin-New York, 1976. |
| [9] |
H. Isozaki,
Some remarks on the multi-dimensional Borg-Levinson theorem, J. Math. Kyoto Univ., 31 (1991), 743-753.
doi: 10.1215/kjm/1250519727. |
| [10] |
A. Katchalov and Y. Kurylev,
Multidimensional inverse problem with incomplete boundary spectral data, Commun. Partial Diff. Eqns., 23 (1998), 55-95.
doi: 10.1080/03605309808821338. |
| [11] |
A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC, Boca Raton, FL, 2001.
doi: 10.1201/9781420036220. |
| [12] |
A. Katchalov, Y. Kurylev, M. Lassas and N. Mandache,
Equivalence of time-domain inverse problems and boundary spectral problem, Inverse problems, 20 (2004), 419-436.
doi: 10.1088/0266-5611/20/2/007. |
| [13] |
O. Kavian, Y. Kian and E. Soccorsi,
Uniqueness and stability results for an inverse spectral problem in a periodic waveguide, Journal de Mathématiques Pures et Appliquées, 104 (2015), 1160-1189.
doi: 10.1016/j.matpur.2015.09.002. |
| [14] |
Y. Kian,
A multidimensional Borg-Levinson theorem for magnetic Schrödinger operators with partial spectral data, J. Spectr. Theory, 8 (2018), 235-269.
doi: 10.4171/JST/195. |
| [15] |
Y. Kian and L. Oksanen, Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations, IMRN, 2017, https://doi.org/10.1093/imrn/rnx263.
doi: 10.1093/imrn/rnx263. |
| [16] |
Y. Kurylev,
An inverse boundary problem for the Schrödinger operator with magnetic field, Journal of Mathematical Physics, 36 (1995), 2761-2776.
doi: 10.1063/1.531064. |
| [17] |
Y. Kurylev and M. Lassas,
Gelf'and inverse problem for a quadratic operator pencil, Journal of Functional Analysis, 176 (2000), 247-263.
doi: 10.1006/jfan.2000.3615. |
| [18] |
Y. Kurylev, L. Oksanen and G. Paternain, Inverse problems for the connection Laplacian, to appear in J. Differential Geom., arXiv: 1509.02645. |
| [19] |
I. Lasiecka, J.-L. Lions and R. Triggiani,
Non homogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192.
|
| [20] |
M. Lassas and L. Oksanen, An inverse problem for a wave equation with sources and observations on disjoint sets, Inverse Problems, 26 (2010), 085012, 19pp.
doi: 10.1088/0266-5611/26/8/085012. |
| [21] |
M. Lassas and L. Oksanen,
Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets, Duke Math. J., 163 (2014), 1071-1103.
|
| [22] |
N. Levinson,
The inverse Strum-Liouville problem, Mat. Tidsskr. B, (1949), 25-30.
|
| [23] |
J.-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Vol. Ⅰ, Dunod, Paris, 1968. |
| [24] |
J.-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Vol. Ⅱ, Dunod, Paris, 1968. |
| [25] |
A. Nachman, J. Sylvester and G. Uhlmann,
An n-dimensional Borg-Levinson theorem, Comm. Math. Phys., 115 (1988), 595-605.
doi: 10.1007/BF01224129. |
| [26] |
L. Päivärinta and V. Serov,
An n-dimensional Borg-Levinson theorem for singular potentials, Adv. in Appl. Math., 29 (2002), 509-520.
|
| [27] |
L. Robbiano and C. Zuily,
Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients, Invent. Math., 131 (1998), 493-539.
doi: 10.1007/s002220050212. |
| [28] |
W. Rudin,
Real and Complex Analysis, McGraw Hill international editions, 1987. |
| [29] |
J. Sylvester and G. Uhlmann,
A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.
doi: 10.2307/1971291. |
| [30] |
D. Tataru,
Unique continuation for solutions to PDE; between Hörmander's theorem and Holmgren's theorem, Commun. Partial Diff. Eqns., 20 (1995), 855-884.
doi: 10.1080/03605309508821117. |
show all references
References:
| [1] |
C. Bardos, G. Lebeau and J. Rauch,
Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM Journal on Control and Optimization, 30 (1992), 1024-1065.
doi: 10.1137/0330055. |
| [2] |
M. Belishev,
An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297 (1987), 524-527.
|
| [3] |
M. Belishev and Y. Kurylev,
To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. Partial Differential Equations, 17 (1992), 767-804.
doi: 10.1080/03605309208820863. |
| [4] |
G. Borg,
Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Acta Math., 78 (1946), 1-96.
doi: 10.1007/BF02421600. |
| [5] |
B. Canuto and O. Kavian, Determining two coefficients in elliptic operators via boundary spectral data: A uniqueness result, Bolletino Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 7 (2004), 207–230. |
| [6] |
M. Choulli and P. Stefanov,
Stability for the multi-dimensional Borg-Levinson theorem with partial spectral data, Commun. Partial Diff. Eqns., 38 (2013), 455-476.
|
| [7] |
I. M. Gel'fand and B. M. Levitan,
On the determination of a differential equation from its spectral function, Izv. Akad. Nauk USSR, Ser. Mat., 15 (1951), 309-360.
|
| [8] |
L. Hörmander, Linear Partial Differential Operators, Springer Verlag, Berlin-New York, 1976. |
| [9] |
H. Isozaki,
Some remarks on the multi-dimensional Borg-Levinson theorem, J. Math. Kyoto Univ., 31 (1991), 743-753.
doi: 10.1215/kjm/1250519727. |
| [10] |
A. Katchalov and Y. Kurylev,
Multidimensional inverse problem with incomplete boundary spectral data, Commun. Partial Diff. Eqns., 23 (1998), 55-95.
doi: 10.1080/03605309808821338. |
| [11] |
A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC, Boca Raton, FL, 2001.
doi: 10.1201/9781420036220. |
| [12] |
A. Katchalov, Y. Kurylev, M. Lassas and N. Mandache,
Equivalence of time-domain inverse problems and boundary spectral problem, Inverse problems, 20 (2004), 419-436.
doi: 10.1088/0266-5611/20/2/007. |
| [13] |
O. Kavian, Y. Kian and E. Soccorsi,
Uniqueness and stability results for an inverse spectral problem in a periodic waveguide, Journal de Mathématiques Pures et Appliquées, 104 (2015), 1160-1189.
doi: 10.1016/j.matpur.2015.09.002. |
| [14] |
Y. Kian,
A multidimensional Borg-Levinson theorem for magnetic Schrödinger operators with partial spectral data, J. Spectr. Theory, 8 (2018), 235-269.
doi: 10.4171/JST/195. |
| [15] |
Y. Kian and L. Oksanen, Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations, IMRN, 2017, https://doi.org/10.1093/imrn/rnx263.
doi: 10.1093/imrn/rnx263. |
| [16] |
Y. Kurylev,
An inverse boundary problem for the Schrödinger operator with magnetic field, Journal of Mathematical Physics, 36 (1995), 2761-2776.
doi: 10.1063/1.531064. |
| [17] |
Y. Kurylev and M. Lassas,
Gelf'and inverse problem for a quadratic operator pencil, Journal of Functional Analysis, 176 (2000), 247-263.
doi: 10.1006/jfan.2000.3615. |
| [18] |
Y. Kurylev, L. Oksanen and G. Paternain, Inverse problems for the connection Laplacian, to appear in J. Differential Geom., arXiv: 1509.02645. |
| [19] |
I. Lasiecka, J.-L. Lions and R. Triggiani,
Non homogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192.
|
| [20] |
M. Lassas and L. Oksanen, An inverse problem for a wave equation with sources and observations on disjoint sets, Inverse Problems, 26 (2010), 085012, 19pp.
doi: 10.1088/0266-5611/26/8/085012. |
| [21] |
M. Lassas and L. Oksanen,
Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets, Duke Math. J., 163 (2014), 1071-1103.
|
| [22] |
N. Levinson,
The inverse Strum-Liouville problem, Mat. Tidsskr. B, (1949), 25-30.
|
| [23] |
J.-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Vol. Ⅰ, Dunod, Paris, 1968. |
| [24] |
J.-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Vol. Ⅱ, Dunod, Paris, 1968. |
| [25] |
A. Nachman, J. Sylvester and G. Uhlmann,
An n-dimensional Borg-Levinson theorem, Comm. Math. Phys., 115 (1988), 595-605.
doi: 10.1007/BF01224129. |
| [26] |
L. Päivärinta and V. Serov,
An n-dimensional Borg-Levinson theorem for singular potentials, Adv. in Appl. Math., 29 (2002), 509-520.
|
| [27] |
L. Robbiano and C. Zuily,
Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients, Invent. Math., 131 (1998), 493-539.
doi: 10.1007/s002220050212. |
| [28] |
W. Rudin,
Real and Complex Analysis, McGraw Hill international editions, 1987. |
| [29] |
J. Sylvester and G. Uhlmann,
A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.
doi: 10.2307/1971291. |
| [30] |
D. Tataru,
Unique continuation for solutions to PDE; between Hörmander's theorem and Holmgren's theorem, Commun. Partial Diff. Eqns., 20 (1995), 855-884.
doi: 10.1080/03605309508821117. |
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