We consider the inverse problem of determining a time-dependent potential q, appearing in the wave equation in with and Ω a bounded domain of , , from partial observations of the solutions on ∂Q. More precisely, we look for observations on ∂Q that allows to recover uniquely a general time-dependent potential q without involving an important set of data. We prove global unique determination of from partial observations on ∂Q. Besides being nonlinear, this problem is related to the inverse problem of determining a semilinear term appearing in a nonlinear hyperbolic equation from boundary measurements.
Mots clés : Inverse problems, Wave equation, Time-dependent potential, Uniqueness, Carleman estimates, Partial data
@article{AIHPC_2017__34_4_973_0, author = {Kian, Yavar}, title = {Unique determination of a time-dependent potential for wave equations from partial data}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {973--990}, publisher = {Elsevier}, volume = {34}, number = {4}, year = {2017}, doi = {10.1016/j.anihpc.2016.07.003}, mrnumber = {3661867}, zbl = {1435.35416}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2016.07.003/} }
TY - JOUR AU - Kian, Yavar TI - Unique determination of a time-dependent potential for wave equations from partial data JO - Annales de l'I.H.P. Analyse non linéaire PY - 2017 SP - 973 EP - 990 VL - 34 IS - 4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2016.07.003/ DO - 10.1016/j.anihpc.2016.07.003 LA - en ID - AIHPC_2017__34_4_973_0 ER -
%0 Journal Article %A Kian, Yavar %T Unique determination of a time-dependent potential for wave equations from partial data %J Annales de l'I.H.P. Analyse non linéaire %D 2017 %P 973-990 %V 34 %N 4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2016.07.003/ %R 10.1016/j.anihpc.2016.07.003 %G en %F AIHPC_2017__34_4_973_0
Kian, Yavar. Unique determination of a time-dependent potential for wave equations from partial data. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 4, pp. 973-990. doi : 10.1016/j.anihpc.2016.07.003. http://www.numdam.org/articles/10.1016/j.anihpc.2016.07.003/
[1] Stability estimate for an inverse wave equation and a multidimensional Borg–Levinson theorem, J. Differ. Equ., Volume 247 (2009) no. 2, pp. 465–494 | MR | Zbl
[2] Lipschitz stability for a hyperbolic inverse problem by finite local boundary data, Appl. Anal., Volume 85 (2006), pp. 1219–1243 | DOI | MR | Zbl
[3] Stability estimate for a hyperbolic inverse problem with time-dependent coefficient, Inverse Probl., Volume 31 (2015), pp. 125010 | MR | Zbl
[4] Recovering a potential from partial Cauchy data, Commun. Partial Differ. Equ., Volume 27 (2002) no. 3–4, pp. 653–668 | MR | Zbl
[5] Une introduction aux problèmes inverses elliptiques et paraboliques, Math. Appl., vol. 65, Springer-Verlag, Berlin, 2009 | MR | Zbl
[6] Stability of the determination of a time-dependent coefficient in parabolic equations, Math. Control Relat. Fields, Volume 3 (2013) no. 2, pp. 143–160 | DOI | MR | Zbl
[7] Determining the time dependent external potential from the DN map in a periodic quantum waveguide, SIAM J. Math. Anal., Volume 47 (2015) no. 6, pp. 4536–4558
[8] Double logarithmic stability estimate in the identification of a scalar potential by a partial elliptic Dirichlet-to-Neumann map, Bulletin SUSU MMCS, Volume 8 (2015) no. 3, pp. 78–94 | Zbl
[9] A new approach to hyperbolic inverse problems, Inverse Probl., Volume 22 (2006) no. 3, pp. 815–831 | DOI | MR | Zbl
[10] Inverse hyperbolic problems with time-dependent coefficients, Commun. Partial Differ. Equ., Volume 32 (2007) no. 11, pp. 1737–1758 | DOI | MR | Zbl
[11] A stability result for a time-dependent potential in a cylindrical domain, Inverse Probl., Volume 29 (2013) no. 6 | DOI | MR | Zbl
[12] The Analysis of Linear Partial Differential Operators, Vol II, Springer-Verlag, Berlin, Heidelberg, 1983 | MR | Zbl
[13] Completness of products of solutions and some inverse problems for PDE, J. Differ. Equ., Volume 92 (1991), pp. 305–316 | DOI | MR | Zbl
[14] An inverse hyperbolic problem with many boundary measurements, Commun. Partial Differ. Equ., Volume 16 (1991), pp. 1183–1195 | MR | Zbl
[15] On uniqueness in inverse problems for semilinear parabolic equations, Arch. Ration. Mech. Anal., Volume 124 (1993), pp. 1–12 | DOI | MR | Zbl
[16] Stability estimates for hyperbolic inverse problems with local boundary data, Inverse Probl., Volume 8 (1992), pp. 193–206 | DOI | MR | Zbl
[17] , IRMA Lect. Math. Theor. Phys., Volume vol. 4, de Gruyter, Berlin (2003), pp. 125–162 | MR | Zbl
[18] The Calderon problem with partial data, Ann. Math., Volume 165 (2007), pp. 567–591 | DOI | MR | Zbl
[19] Stability of the determination of a coefficient for wave equations in an infinite waveguide, Inverse Probl. Imaging, Volume 8 (2014) no. 3, pp. 713–732 | MR | Zbl
[20] Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations (preprint) | arXiv | MR
[21] Non homogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., Volume 65 (1986), pp. 149–192 | MR | Zbl
[22] Problèmes aux limites non homogènes et applications, Vol. I, Dunod, Paris, 1968 | MR | Zbl
[23] Problèmes aux limites non homogènes et applications, Vol. II, Dunod, Paris, 1968 | MR | Zbl
[24] Stable determination of a simple metric, a co-vector field and a potential from the hyperbolic Dirichlet-to-Neumann map, Commun. Partial Differ. Equ., Volume 39 (2014), pp. 120–145 | DOI | MR | Zbl
[25] Reconstruction in the Calderón problem with partial data, Commun. Partial Differ. Equ., Volume 35 (2010), pp. 375–390 | DOI | MR | Zbl
[26] Property C and an inverse problem for a hyperbolic equation, J. Math. Anal. Appl., Volume 156 (1991), pp. 209–219 | MR | Zbl
[27] Uniqueness for an inverse problem for the wave equation, Commun. Partial Differ. Equ., Volume 13 (1988) no. 1, pp. 87–96 | DOI | MR | Zbl
[28] An inverse problem of the wave equation, Math. Z., Volume 206 (1991), pp. 119–130 | MR | Zbl
[29] Determination of time-dependent coefficients for a hyperbolic inverse problem, Inverse Probl., Volume 29 (2013) no. 9 | DOI | MR | Zbl
[30] Uniqueness of the multi-dimensional inverse scattering problem for time dependent potentials, Math. Z., Volume 201 (1989) no. 4, pp. 541–559 | DOI | MR | Zbl
[31] Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media, J. Funct. Anal., Volume 154 (1998), pp. 330–358 | DOI | MR | Zbl
[32] Stable determination of the hyperbolic Dirichlet-to-Neumann map for generic simple metrics, Int. Math. Res. Not. (IMRN), Volume 17 (2005), pp. 1047–1061 | MR | Zbl
[33] Unique continuation for solutions to PDE; between Hörmander's theorem and Holmgren's theorem, Commun. Partial Differ. Equ., Volume 20 (1995), pp. 855–884 | MR | Zbl
[34] Stable determination of X-ray transforms of time dependent potentials from partial boundary data, Commun. Partial Differ. Equ., Volume 39 (2014), pp. 2169–2197 | DOI | MR | Zbl
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