Application of the boundary control method to partial data Borg-Levinson inverse spectral problem

Yavar Kian; Morgan Morancey; Lauri Oksanen

doi:10.3934/mcrf.2019015

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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

Yavar Kian ^1, , Morgan Morancey ^2,, and Lauri Oksanen ^3,

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

* Corresponding author

Received April 2017 Revised January 2018 Published November 2018

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 & 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. Google Scholar
[2]	M. Belishev, An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297 (1987), 524-527. Google Scholar
[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. Google Scholar
[4]	G. Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Acta Math., 78 (1946), 1-96. doi: 10.1007/BF02421600. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[8]	L. Hörmander, Linear Partial Differential Operators, Springer Verlag, Berlin-New York, 1976. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420036220. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	Y. Kurylev, L. Oksanen and G. Paternain, Inverse problems for the connection Laplacian, to appear in J. Differential Geom., arXiv: 1509.02645. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[22]	N. Levinson, The inverse Strum-Liouville problem, Mat. Tidsskr. B, (1949), 25-30. Google Scholar
[23]	J.-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Vol. Ⅰ, Dunod, Paris, 1968. Google Scholar
[24]	J.-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Vol. Ⅱ, Dunod, Paris, 1968. Google Scholar
[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. Google Scholar
[26]	L. Päivärinta and V. Serov, An n-dimensional Borg-Levinson theorem for singular potentials, Adv. in Appl. Math., 29 (2002), 509-520. Google Scholar
[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. Google Scholar
[28]	W. Rudin, Real and Complex Analysis, McGraw Hill international editions, 1987. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[2]	M. Belishev, An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297 (1987), 524-527. Google Scholar
[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. Google Scholar
[4]	G. Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Acta Math., 78 (1946), 1-96. doi: 10.1007/BF02421600. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[8]	L. Hörmander, Linear Partial Differential Operators, Springer Verlag, Berlin-New York, 1976. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420036220. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	Y. Kurylev, L. Oksanen and G. Paternain, Inverse problems for the connection Laplacian, to appear in J. Differential Geom., arXiv: 1509.02645. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[22]	N. Levinson, The inverse Strum-Liouville problem, Mat. Tidsskr. B, (1949), 25-30. Google Scholar
[23]	J.-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Vol. Ⅰ, Dunod, Paris, 1968. Google Scholar
[24]	J.-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Vol. Ⅱ, Dunod, Paris, 1968. Google Scholar
[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. Google Scholar
[26]	L. Päivärinta and V. Serov, An n-dimensional Borg-Levinson theorem for singular potentials, Adv. in Appl. Math., 29 (2002), 509-520. Google Scholar
[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. Google Scholar
[28]	W. Rudin, Real and Complex Analysis, McGraw Hill international editions, 1987. Google Scholar
[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. Google Scholar
[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. Google Scholar

Figure 1. Geometric condition (3.10)

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Figure 2. Sets $A_{x, \varepsilon}$

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Figure 3. Support of the geometric optics solution

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Application of the boundary control method to partial data Borg-Levinson inverse spectral problem

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