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Uniqueness on recovery of piecewise constant conductivity and inner core with one measurement
Determination of singular time-dependent coefficients for wave equations from full and partial data
1. | Beijing Computational Science Research Center, Building 9, East Zone, ZPark Ⅱ, No.10 Xibeiwang East Road, Haidian District, Beijing 100193, China |
2. | Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France |
We study the problem of determining uniquely a time-dependent singular potential $q$, appearing in the wave equation $\partial_t^2u-Δ_x u+q(t,x)u = 0$ in $Q = (0,T)×Ω$ with $T>0$ and $Ω$ a $ \mathcal C^2$ bounded domain of $\mathbb{R}^n$, $n≥2$. We start by considering the unique determination of some general singular time-dependent coefficients. Then, by weakening the singularities of the set of admissible coefficients, we manage to reduce the set of data that still guaranties unique recovery of such a coefficient. To our best knowledge, this paper is the first claiming unique determination of unbounded time-dependent coefficients, which is motivated by the problem of determining general nonlinear terms appearing in nonlinear wave equations.
References:
[1] |
K. Astala and L. Päivärinta,
Calderón's inverse conductivity problem in the plane, Ann. of Math., 163 (2006), 265-299.
doi: 10.4007/annals.2006.163.265. |
[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] |
M. Bellassoued and I. Ben Aicha,
Stable determination outside a cloaking region of two time-dependent coefficients in an hyperbolic equation from Dirichlet to Neumann map, Jour. Math. Anal. Appl., 449 (2017), 46-76.
doi: 10.1016/j.jmaa.2016.11.082. |
[5] |
M. Bellassoued, M. Choulli and M. Yamamoto,
Stability estimate for an inverse wave equation and a multidimensional Borg-Levinson theorem, J. Diff. Equat., 247 (2009), 465-494.
doi: 10.1016/j.jde.2009.03.024. |
[6] |
M. Bellassoued and D. Dos Santos Ferreira,
Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map, Inverse Probl. Imaging, 5 (2011), 745-773.
doi: 10.3934/ipi.2011.5.745. |
[7] |
M. Bellassoued, D. Jellali and M. Yamamoto,
Lipschitz stability for a hyperbolic inverse problem by finite local boundary data, Appl. Anal., 85 (2006), 1219-1243.
doi: 10.1080/00036810600787873. |
[8] |
I. Ben Aicha, Stability estimate for hyperbolic inverse problem with time-dependent coefficient, Inverse Problems, 31 (2015), 125010, 21pp. |
[9] |
A. Bukhgeim and M. Klibanov,
Global uniqueness of a class of multidimensional inverse problem, Sov. Math.-Dokl., 24 (1981), 244-247.
|
[10] |
A. L. Bukhgeim and G. Uhlmann,
Recovering a potential from partial Cauchy data, Commun. Partial Diff. Eqns., 27 (2002), 653-668.
doi: 10.1081/PDE-120002868. |
[11] |
P. Caro and K. M. Rogers, Global Uniqueness for The Calderón Problem with Lipschitz Conductivities, Forum of Mathematics, Pi, 4 (2016), e2, 28 pp. |
[12] |
M. Choulli and Y. Kian,
Stability of the determination of a time-dependent coefficient in parabolic equations, MCRF, 3 (2013), 143-160.
doi: 10.3934/mcrf.2013.3.143. |
[13] |
M. Choulli and Y. Kian, Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term, J. Math. Pures Appl. (2017).
doi: 10.1016/j.matpur.2017.12.003. |
[14] |
M. Choulli, Y. Kian and E. Soccorsi,
Determining the time dependent external potential from the DN map in a periodic quantum waveguide, SIAM J. Math. Anal., 47 (2015), 4536-4558.
doi: 10.1137/140986268. |
[15] |
M. Choulli, Y. Kian and E. Soccorsi, Stability result for elliptic inverse periodic coefficient problem by partial Dirichlet-to-Neumann map, to appear Journal of Spectral Theory, arXiv: 1601.05355. |
[16] |
M. Choulli, Y. Kian and E. Soccorsi,
On the Calderón problem in periodic cylindrical domain with partial Dirichlet and Neumann data, Mathematical Methods in the Applied Sciences, 40 (2017), 5959-5974.
doi: 10.1002/mma.4446. |
[17] |
F. Chung and L. Tzou, The $L^p$ Carleman estimate and a partial data inverse problem, preprint, arXiv: 1610.01715. |
[18] |
G. Eskin,
A new approach to hyperbolic inverse problems, Inverse Problems, 22 (2006), 815-831.
doi: 10.1088/0266-5611/22/3/005. |
[19] |
G. Eskin,
Inverse hyperbolic problems with time-dependent coefficients, Commun. Partial Diff. Eqns., 32 (2007), 1737-1758.
doi: 10.1080/03605300701382340. |
[20] |
G. Eskin,
Inverse problems for general second order hyperbolic equations with time-dependent coefficients, Bull. Math. Sci., 7 (2017), 247-307.
doi: 10.1007/s13373-017-0100-2. |
[21] |
D. D. Feirrera, C. Kenig and M. Salo,
Determining an unbounded potential from Cauchy data in admissible geometries, Comm. Partial Differential Equations, 38 (2013), 50-68.
doi: 10.1080/03605302.2012.736911. |
[22] |
K. Fujishiro and Y. Kian,
Determination of time dependent factors of coefficients in fractional diffusion equations, MCRF, 6 (2016), 251-269.
doi: 10.3934/mcrf.2016003. |
[23] |
P. Gaitan and Y. Kian, A stability result for a time-dependent potential in a cylindrical domain, Inverse Problems, 29 (2013), 065006, 18pp. |
[24] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985. |
[25] |
B. Haberman,
Uniqueness in Calderón's problem for conductivities with unbounded gradient, Comm. Math. Phys., 340 (2015), 639-659.
doi: 10.1007/s00220-015-2460-3. |
[26] |
B. Haberman and D. Tataru,
Uniqueness in Calderón's problem with Lipschitz conductivities, Duke Math. Journal, 162 (2013), 496-516.
|
[27] |
L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol Ⅲ, Springer-Verlag, Berlin, Heidelberg, 1983. |
[28] |
V. Isakov,
Completness of products of solutions and some inverse problems for PDE, J. Diff. Equat., 92 (1991), 305-316.
doi: 10.1016/0022-0396(91)90051-A. |
[29] |
V. Isakov,
An inverse hyperbolic problem with many boundary measurements, Commun. Partial Diff. Eqns., 16 (1991), 1183-1195.
doi: 10.1080/03605309108820794. |
[30] |
V. Isakov,
On uniqueness in inverse problems for semilinear parabolic equations, Arch. Rat. Mech. Anal., 124 (1993), 1-12.
doi: 10.1007/BF00392201. |
[31] |
A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 123 Chapman & Hall/CRC, Boca Raton, FL, 2001. |
[32] |
O. Kavian, Four lectures on parameter identification, three courses on partial differential equations, IRMA Lect. Math. Theor. Phys., de Gruyter, Berlin, 4 (2003), 125-162. |
[33] |
C. E. Kenig, J. Sjöstrand and G. Uhlmann,
The Calderon problem with partial data, Ann. of Math., 165 (2007), 567-591.
doi: 10.4007/annals.2007.165.567. |
[34] |
Y. Kian,
Stability of the determination of a coefficient for wave equations in an infinite waveguide, Inverse Probl. Imaging, 8 (2014), 713-732.
doi: 10.3934/ipi.2014.8.713. |
[35] |
Y. Kian,
Stability in the determination of a time-dependent coefficient for wave equations from partial data, Jour. Math. Anal. Appl., 436 (2016), 408-428.
doi: 10.1016/j.jmaa.2015.12.018. |
[36] |
Y. Kian,
Unique determination of a time-dependent potential for wave equations from partial data, Annales de l'IHP (C) Nonlinear Analysis, 34 (2017), 973-990.
doi: 10.1016/j.anihpc.2016.07.003. |
[37] |
Y. Kian,
Recovery of time-dependent damping coefficients and potentials appearing in wave equations from partial data, SIAM J. Math. Anal., 48 (2016), 4021-4046.
doi: 10.1137/16M1076708. |
[38] |
Y. Kian and L. Oksanen, Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations, to appear in International Math Research Notices, available at https://doi.org/10.1093/imrn/rnx263. |
[39] |
Y. Kian, L. Oksanen and M. Morancey, Application of the boundary control method to partial data Borg-Levinson inverse spectral problem, preprint, arXiv: 1703.08832. |
[40] |
Y. Kian and E. Soccorsi, Hölder stably determining the time-dependent electromagnetic potential of the Schrödinger equation, preprint, arXiv: 1705.01322. |
[41] |
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.
|
[42] |
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. |
[43] |
J. -L. Lions and E. Magenes, Problémes aux Limites non Homogénes et Applications, Vol. Ⅰ, Dunod, Paris, 1968. |
[44] |
Rakesh and A. G. Ramm,
Property C and an inverse problem for a hyperbolic equation, J. Math. Anal. Appl., 156 (1991), 209-219.
doi: 10.1016/0022-247X(91)90391-C. |
[45] |
Rakesh and W. Symes,
Uniqueness for an inverse problem for the wave equation, Commun. Partial Diff. Eqns., 13 (1988), 87-96.
doi: 10.1080/03605308808820539. |
[46] |
A. G. Ramm and J. Sjöstrand,
An inverse problem of the wave equation, Math. Z., 206 (1991), 119-130.
doi: 10.1007/BF02571330. |
[47] |
R. Salazar, Determination of time-dependent coefficients for a hyperbolic inverse problem, Inverse Problems, 29 (2013), 095015, 17pp. |
[48] |
P. Stefanov,
Uniqueness of the multi-dimensional inverse scattering problem for time dependent potentials, Math. Z., 201 (1989), 541-559.
doi: 10.1007/BF01215158. |
[49] |
P. Stefanov and G. Uhlmann,
Stable determination of the hyperbolic Dirichlet-to-Neumann map for generic simple metrics, International Math Research Notices, 17 (2005), 1047-1061.
|
[50] |
P. Stefanov and Y. Yang, The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds, preprint, arXiv: 1607.08690. |
[51] |
A. Waters,
Stable determination of X-ray transforms of time dependent potentials from partial boundary data, Commun. Partial Diff. Eqns., 39 (2014), 2169-2197.
doi: 10.1080/03605302.2014.930486. |
show all references
References:
[1] |
K. Astala and L. Päivärinta,
Calderón's inverse conductivity problem in the plane, Ann. of Math., 163 (2006), 265-299.
doi: 10.4007/annals.2006.163.265. |
[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] |
M. Bellassoued and I. Ben Aicha,
Stable determination outside a cloaking region of two time-dependent coefficients in an hyperbolic equation from Dirichlet to Neumann map, Jour. Math. Anal. Appl., 449 (2017), 46-76.
doi: 10.1016/j.jmaa.2016.11.082. |
[5] |
M. Bellassoued, M. Choulli and M. Yamamoto,
Stability estimate for an inverse wave equation and a multidimensional Borg-Levinson theorem, J. Diff. Equat., 247 (2009), 465-494.
doi: 10.1016/j.jde.2009.03.024. |
[6] |
M. Bellassoued and D. Dos Santos Ferreira,
Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map, Inverse Probl. Imaging, 5 (2011), 745-773.
doi: 10.3934/ipi.2011.5.745. |
[7] |
M. Bellassoued, D. Jellali and M. Yamamoto,
Lipschitz stability for a hyperbolic inverse problem by finite local boundary data, Appl. Anal., 85 (2006), 1219-1243.
doi: 10.1080/00036810600787873. |
[8] |
I. Ben Aicha, Stability estimate for hyperbolic inverse problem with time-dependent coefficient, Inverse Problems, 31 (2015), 125010, 21pp. |
[9] |
A. Bukhgeim and M. Klibanov,
Global uniqueness of a class of multidimensional inverse problem, Sov. Math.-Dokl., 24 (1981), 244-247.
|
[10] |
A. L. Bukhgeim and G. Uhlmann,
Recovering a potential from partial Cauchy data, Commun. Partial Diff. Eqns., 27 (2002), 653-668.
doi: 10.1081/PDE-120002868. |
[11] |
P. Caro and K. M. Rogers, Global Uniqueness for The Calderón Problem with Lipschitz Conductivities, Forum of Mathematics, Pi, 4 (2016), e2, 28 pp. |
[12] |
M. Choulli and Y. Kian,
Stability of the determination of a time-dependent coefficient in parabolic equations, MCRF, 3 (2013), 143-160.
doi: 10.3934/mcrf.2013.3.143. |
[13] |
M. Choulli and Y. Kian, Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term, J. Math. Pures Appl. (2017).
doi: 10.1016/j.matpur.2017.12.003. |
[14] |
M. Choulli, Y. Kian and E. Soccorsi,
Determining the time dependent external potential from the DN map in a periodic quantum waveguide, SIAM J. Math. Anal., 47 (2015), 4536-4558.
doi: 10.1137/140986268. |
[15] |
M. Choulli, Y. Kian and E. Soccorsi, Stability result for elliptic inverse periodic coefficient problem by partial Dirichlet-to-Neumann map, to appear Journal of Spectral Theory, arXiv: 1601.05355. |
[16] |
M. Choulli, Y. Kian and E. Soccorsi,
On the Calderón problem in periodic cylindrical domain with partial Dirichlet and Neumann data, Mathematical Methods in the Applied Sciences, 40 (2017), 5959-5974.
doi: 10.1002/mma.4446. |
[17] |
F. Chung and L. Tzou, The $L^p$ Carleman estimate and a partial data inverse problem, preprint, arXiv: 1610.01715. |
[18] |
G. Eskin,
A new approach to hyperbolic inverse problems, Inverse Problems, 22 (2006), 815-831.
doi: 10.1088/0266-5611/22/3/005. |
[19] |
G. Eskin,
Inverse hyperbolic problems with time-dependent coefficients, Commun. Partial Diff. Eqns., 32 (2007), 1737-1758.
doi: 10.1080/03605300701382340. |
[20] |
G. Eskin,
Inverse problems for general second order hyperbolic equations with time-dependent coefficients, Bull. Math. Sci., 7 (2017), 247-307.
doi: 10.1007/s13373-017-0100-2. |
[21] |
D. D. Feirrera, C. Kenig and M. Salo,
Determining an unbounded potential from Cauchy data in admissible geometries, Comm. Partial Differential Equations, 38 (2013), 50-68.
doi: 10.1080/03605302.2012.736911. |
[22] |
K. Fujishiro and Y. Kian,
Determination of time dependent factors of coefficients in fractional diffusion equations, MCRF, 6 (2016), 251-269.
doi: 10.3934/mcrf.2016003. |
[23] |
P. Gaitan and Y. Kian, A stability result for a time-dependent potential in a cylindrical domain, Inverse Problems, 29 (2013), 065006, 18pp. |
[24] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985. |
[25] |
B. Haberman,
Uniqueness in Calderón's problem for conductivities with unbounded gradient, Comm. Math. Phys., 340 (2015), 639-659.
doi: 10.1007/s00220-015-2460-3. |
[26] |
B. Haberman and D. Tataru,
Uniqueness in Calderón's problem with Lipschitz conductivities, Duke Math. Journal, 162 (2013), 496-516.
|
[27] |
L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol Ⅲ, Springer-Verlag, Berlin, Heidelberg, 1983. |
[28] |
V. Isakov,
Completness of products of solutions and some inverse problems for PDE, J. Diff. Equat., 92 (1991), 305-316.
doi: 10.1016/0022-0396(91)90051-A. |
[29] |
V. Isakov,
An inverse hyperbolic problem with many boundary measurements, Commun. Partial Diff. Eqns., 16 (1991), 1183-1195.
doi: 10.1080/03605309108820794. |
[30] |
V. Isakov,
On uniqueness in inverse problems for semilinear parabolic equations, Arch. Rat. Mech. Anal., 124 (1993), 1-12.
doi: 10.1007/BF00392201. |
[31] |
A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 123 Chapman & Hall/CRC, Boca Raton, FL, 2001. |
[32] |
O. Kavian, Four lectures on parameter identification, three courses on partial differential equations, IRMA Lect. Math. Theor. Phys., de Gruyter, Berlin, 4 (2003), 125-162. |
[33] |
C. E. Kenig, J. Sjöstrand and G. Uhlmann,
The Calderon problem with partial data, Ann. of Math., 165 (2007), 567-591.
doi: 10.4007/annals.2007.165.567. |
[34] |
Y. Kian,
Stability of the determination of a coefficient for wave equations in an infinite waveguide, Inverse Probl. Imaging, 8 (2014), 713-732.
doi: 10.3934/ipi.2014.8.713. |
[35] |
Y. Kian,
Stability in the determination of a time-dependent coefficient for wave equations from partial data, Jour. Math. Anal. Appl., 436 (2016), 408-428.
doi: 10.1016/j.jmaa.2015.12.018. |
[36] |
Y. Kian,
Unique determination of a time-dependent potential for wave equations from partial data, Annales de l'IHP (C) Nonlinear Analysis, 34 (2017), 973-990.
doi: 10.1016/j.anihpc.2016.07.003. |
[37] |
Y. Kian,
Recovery of time-dependent damping coefficients and potentials appearing in wave equations from partial data, SIAM J. Math. Anal., 48 (2016), 4021-4046.
doi: 10.1137/16M1076708. |
[38] |
Y. Kian and L. Oksanen, Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations, to appear in International Math Research Notices, available at https://doi.org/10.1093/imrn/rnx263. |
[39] |
Y. Kian, L. Oksanen and M. Morancey, Application of the boundary control method to partial data Borg-Levinson inverse spectral problem, preprint, arXiv: 1703.08832. |
[40] |
Y. Kian and E. Soccorsi, Hölder stably determining the time-dependent electromagnetic potential of the Schrödinger equation, preprint, arXiv: 1705.01322. |
[41] |
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.
|
[42] |
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. |
[43] |
J. -L. Lions and E. Magenes, Problémes aux Limites non Homogénes et Applications, Vol. Ⅰ, Dunod, Paris, 1968. |
[44] |
Rakesh and A. G. Ramm,
Property C and an inverse problem for a hyperbolic equation, J. Math. Anal. Appl., 156 (1991), 209-219.
doi: 10.1016/0022-247X(91)90391-C. |
[45] |
Rakesh and W. Symes,
Uniqueness for an inverse problem for the wave equation, Commun. Partial Diff. Eqns., 13 (1988), 87-96.
doi: 10.1080/03605308808820539. |
[46] |
A. G. Ramm and J. Sjöstrand,
An inverse problem of the wave equation, Math. Z., 206 (1991), 119-130.
doi: 10.1007/BF02571330. |
[47] |
R. Salazar, Determination of time-dependent coefficients for a hyperbolic inverse problem, Inverse Problems, 29 (2013), 095015, 17pp. |
[48] |
P. Stefanov,
Uniqueness of the multi-dimensional inverse scattering problem for time dependent potentials, Math. Z., 201 (1989), 541-559.
doi: 10.1007/BF01215158. |
[49] |
P. Stefanov and G. Uhlmann,
Stable determination of the hyperbolic Dirichlet-to-Neumann map for generic simple metrics, International Math Research Notices, 17 (2005), 1047-1061.
|
[50] |
P. Stefanov and Y. Yang, The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds, preprint, arXiv: 1607.08690. |
[51] |
A. Waters,
Stable determination of X-ray transforms of time dependent potentials from partial boundary data, Commun. Partial Diff. Eqns., 39 (2014), 2169-2197.
doi: 10.1080/03605302.2014.930486. |
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