# American Institute of Mathematical Sciences

February  2019, 13(1): 81-91. doi: 10.3934/ipi.2019005

## Recovering two coefficients in an elliptic equation via phaseless information

 1 Sobolev Institute of Mathematics, Siberian Division of Russian Academy of Sciences, Acad. Koptyug prospekt 4, 630090 Novosibirsk, Russia 2 Department of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153, Japan 3 Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

Received  January 2018 Revised  April 2018 Published  December 2018

For fixed $y \in \mathbb{R}^3$, we consider the equation $L u+k^2u = - δ(x-y), \>x \in \mathbb{R}^3$, where $L=\text{div}(n(x)^{-2}\nabla)+q(x)$, $k >0$ is a frequency, $n(x)$ is a refraction index and $q(x)$ is a potential. Assuming that the refraction index $n(x)$ is different from $1$ only inside a bounded compact domain $Ω$ with a smooth boundary $S$ and the potential $q(x)$ vanishes outside of the same domain, we study an inverse problem of finding both coefficients inside $Ω$ from some given information on solutions of the elliptic equation. Namely, it is supposed that the point source located at point $y \in S$ is a variable parameter of the problem. Then for the solution $u(x,y,k)$ of the above equation satisfying the radiation condition, we assume to be given the following phaseless information $f(x,y,k)=|u(x,y,k)|^2$ for all $x,y \in S$ and for all $k≥ k_0>0$, where $k_0$ is some constant. We prove that this phaseless information uniquely determines both coefficients $n(x)$ and $q(x)$ inside $Ω$.

Citation: Vladimir G. Romanov, Masahiro Yamamoto. Recovering two coefficients in an elliptic equation via phaseless information. Inverse Problems & Imaging, 2019, 13 (1) : 81-91. doi: 10.3934/ipi.2019005
##### References:
 [1] J. N. Bernstein and M. L. Gerver, On the problem of integral geometry for a family of geodesics and the inverse kinematics seismic problem, Dokl. Akad. Nauk SSSR, 243 (1978), 302-305 (in Russian). [2] K. Chadan and P. S. Sabatier, Inverse Problems in Quantum Scattering Theory, Texts and Monographs in Physics, Springer-Verlag, New York - Berlin, 1977. [3] M. V. Klibanov, Phaseless inverse scattering problems in three dimensions, SIAM J. Appl. Math., 74 (2014), 392-410. doi: 10.1137/130926250. [4] M. V. Klibanov, On the first solution of a long standing problem: Uniqueness of the phaseless quantum inverse scattering problem in 3-d, Applied Mathematics Letters, 37 (2014), 82-85. doi: 10.1016/j.aml.2014.06.005. [5] M. V. Klibanov, Uniqueness of two phaseless non-overdetermined inverse acoustics problems in 3-d, Applicable Analysis, 93 (2014), 1135-1149. doi: 10.1080/00036811.2013.818136. [6] M. V. Klibanov, A phaseless inverse scattering problem for the 3-D Helmholtz equation, Inverse Problems and Imaging, 11 (2017), 263-276. doi: 10.3934/ipi.2017013. [7] M. V. Klibanov and V. G. Romanov, The first solution of a long standing problem: Reconstruction formula for a 3-d phaseless inverse scattering problem for the Schrödinger equation, J. Inverse and Ill-Posed Problems, 23 (2015), 415-428. doi: 10.1515/jiip-2015-0025. [8] M. V. Klibanov and V. G. Romanov, Explicit solution of 3-D phaseless inverse scattering problems for the Schrödinger equation: The plane wave case, Eurasian J. of Math. and Comp. Appl., 3 (2015), 48-63. [9] M. V. Klibanov and V. G. Romanov, Explicit formula for the solution of the phaseless inverse scattering problem of imaging of nano structures, J. of Inverse and Ill-Posed Problems, 23 (2015), 187-193. doi: 10.1515/jiip-2015-0004. [10] M. V. Klibanov and V. G. Romanov, Two reconstruction procedures for a 3-d phaseless inverse scattering problem for the generalized Helmholtz equation, Inverse Problems, 32 (2016), 015005 (16pp). doi: 10.1088/0266-5611/32/1/015005. [11] M. V. Klibanov and V. G. Romanov, Reconstruction procedures for two inverse scattering problem without the phase information, SIAM J. Appl. Math., 76 (2016), 178-196. doi: 10.1137/15M1022367. [12] M. V. Klibanov and V. G. Romanov, Uniqueness of a 3-D coefficient inverse scattering problem without the phase information, Inverse Problems, 33 (2017), 095007 (10 pp), https://doi.org/10.1088/1361-6420/aa7a18 doi: 10.1088/1361-6420/aa7a18. [13] M. M. Lavrent'ev and V. G. Romanov, On three linearized inverse problems for hyperbolic equations, Soviet Math. Dokl., 7 (1966), 1650-1652. [14] M. M. Lavrent'ev, V. G. Romanov and S. P. Shishat·skiǐ, Ill-Posed Problems of Mathematical Physics and Analysis, Transl. of Math. Monographs, Vol. 64, AMS, Providence, Rhode Island, 1986. [15] R. G. Mukhometov, The reconstruction problem of a two-dimensional Riemannian metric and integral geometry, Dokl. Akad. Nauk SSSR, 232 (1977), 32-35. [16] R. G. Mukhometov and V. G. Romanov, On the problem of determining an isotropic Riemannian metric in $n$-dimensional space, Dokl. Akad. Nauk SSSR, 243 (1978), 41-44. [17] R. G. Newton, Inverse Schrödinger Scattering in Three Dimensions, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-83671-8. [18] R. G. Novikov, Formulas for phase recovering from phaseless scattering data at fixed frequency, Bulletin des Sciences Mathématiques, 139 (2015), 923-936. doi: 10.1016/j.bulsci.2015.04.005. [19] R. G. Novikov, Phaseless inverse scattering in the one-dimensional case, Eurasian J. of Math. and Comp. Appl., 3 (2015), 64-70. [20] R. G. Novikov, Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions, J. Geometrical Analysis, 26 (2016), 346-359. doi: 10.1007/s12220-014-9553-7. [21] V. G. Romanov, Reconstructing a function by means of integrals along a family of curves, Siberian Math. J., 8 (1967), 923-925. [22] V. G. Romanov, Integral Geometry and Inverse Problems for Hyperbolic Equations, Springer-Verlag, Springers Tracts in Natural Philosophy, Vol. 26, Berlin, 1974. [23] V. G. Romanov, Integral geometry on geodesics of the isotropic Riemannian metric, Soviet Math. Dokl., 19 (1978), 847-851. [24] V. G. Romanov, Inverse Problems of Mathematical Physics, VNU Science Press, Utrecht, 1987. [25] V. G. Romanov, Investigation Methods for Inverse Problems, VSP, Utrecht, 2002. doi: 10.1515/9783110943849. [26] V. G. Romanov, The problem of recovering the permittivity coefficient from the modulus of the scattered electromagnetic field, Siberian Math. J., 58 (2017), 711-717. doi: 10.1134/s0037446617040176. [27] V. G. Romanov, Problem of determining the permittivity in the stationary system of Maxwell equations, Doklady Math., 95 (2017), 230-234. doi: 10.1134/s1064562417030164. [28] B. R. Vainberg Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach Science Publishers, New York, 1989.

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##### References:
 [1] J. N. Bernstein and M. L. Gerver, On the problem of integral geometry for a family of geodesics and the inverse kinematics seismic problem, Dokl. Akad. Nauk SSSR, 243 (1978), 302-305 (in Russian). [2] K. Chadan and P. S. Sabatier, Inverse Problems in Quantum Scattering Theory, Texts and Monographs in Physics, Springer-Verlag, New York - Berlin, 1977. [3] M. V. Klibanov, Phaseless inverse scattering problems in three dimensions, SIAM J. Appl. Math., 74 (2014), 392-410. doi: 10.1137/130926250. [4] M. V. Klibanov, On the first solution of a long standing problem: Uniqueness of the phaseless quantum inverse scattering problem in 3-d, Applied Mathematics Letters, 37 (2014), 82-85. doi: 10.1016/j.aml.2014.06.005. [5] M. V. Klibanov, Uniqueness of two phaseless non-overdetermined inverse acoustics problems in 3-d, Applicable Analysis, 93 (2014), 1135-1149. doi: 10.1080/00036811.2013.818136. [6] M. V. Klibanov, A phaseless inverse scattering problem for the 3-D Helmholtz equation, Inverse Problems and Imaging, 11 (2017), 263-276. doi: 10.3934/ipi.2017013. [7] M. V. Klibanov and V. G. Romanov, The first solution of a long standing problem: Reconstruction formula for a 3-d phaseless inverse scattering problem for the Schrödinger equation, J. Inverse and Ill-Posed Problems, 23 (2015), 415-428. doi: 10.1515/jiip-2015-0025. [8] M. V. Klibanov and V. G. Romanov, Explicit solution of 3-D phaseless inverse scattering problems for the Schrödinger equation: The plane wave case, Eurasian J. of Math. and Comp. Appl., 3 (2015), 48-63. [9] M. V. Klibanov and V. G. Romanov, Explicit formula for the solution of the phaseless inverse scattering problem of imaging of nano structures, J. of Inverse and Ill-Posed Problems, 23 (2015), 187-193. doi: 10.1515/jiip-2015-0004. [10] M. V. Klibanov and V. G. Romanov, Two reconstruction procedures for a 3-d phaseless inverse scattering problem for the generalized Helmholtz equation, Inverse Problems, 32 (2016), 015005 (16pp). doi: 10.1088/0266-5611/32/1/015005. [11] M. V. Klibanov and V. G. Romanov, Reconstruction procedures for two inverse scattering problem without the phase information, SIAM J. Appl. Math., 76 (2016), 178-196. doi: 10.1137/15M1022367. [12] M. V. Klibanov and V. G. Romanov, Uniqueness of a 3-D coefficient inverse scattering problem without the phase information, Inverse Problems, 33 (2017), 095007 (10 pp), https://doi.org/10.1088/1361-6420/aa7a18 doi: 10.1088/1361-6420/aa7a18. [13] M. M. Lavrent'ev and V. G. Romanov, On three linearized inverse problems for hyperbolic equations, Soviet Math. Dokl., 7 (1966), 1650-1652. [14] M. M. Lavrent'ev, V. G. Romanov and S. P. Shishat·skiǐ, Ill-Posed Problems of Mathematical Physics and Analysis, Transl. of Math. Monographs, Vol. 64, AMS, Providence, Rhode Island, 1986. [15] R. G. Mukhometov, The reconstruction problem of a two-dimensional Riemannian metric and integral geometry, Dokl. Akad. Nauk SSSR, 232 (1977), 32-35. [16] R. G. Mukhometov and V. G. Romanov, On the problem of determining an isotropic Riemannian metric in $n$-dimensional space, Dokl. Akad. Nauk SSSR, 243 (1978), 41-44. [17] R. G. Newton, Inverse Schrödinger Scattering in Three Dimensions, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-83671-8. [18] R. G. Novikov, Formulas for phase recovering from phaseless scattering data at fixed frequency, Bulletin des Sciences Mathématiques, 139 (2015), 923-936. doi: 10.1016/j.bulsci.2015.04.005. [19] R. G. Novikov, Phaseless inverse scattering in the one-dimensional case, Eurasian J. of Math. and Comp. Appl., 3 (2015), 64-70. [20] R. G. Novikov, Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions, J. Geometrical Analysis, 26 (2016), 346-359. doi: 10.1007/s12220-014-9553-7. [21] V. G. Romanov, Reconstructing a function by means of integrals along a family of curves, Siberian Math. J., 8 (1967), 923-925. [22] V. G. Romanov, Integral Geometry and Inverse Problems for Hyperbolic Equations, Springer-Verlag, Springers Tracts in Natural Philosophy, Vol. 26, Berlin, 1974. [23] V. G. Romanov, Integral geometry on geodesics of the isotropic Riemannian metric, Soviet Math. Dokl., 19 (1978), 847-851. [24] V. G. Romanov, Inverse Problems of Mathematical Physics, VNU Science Press, Utrecht, 1987. [25] V. G. Romanov, Investigation Methods for Inverse Problems, VSP, Utrecht, 2002. doi: 10.1515/9783110943849. [26] V. G. Romanov, The problem of recovering the permittivity coefficient from the modulus of the scattered electromagnetic field, Siberian Math. J., 58 (2017), 711-717. doi: 10.1134/s0037446617040176. [27] V. G. Romanov, Problem of determining the permittivity in the stationary system of Maxwell equations, Doklady Math., 95 (2017), 230-234. doi: 10.1134/s1064562417030164. [28] B. R. Vainberg Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach Science Publishers, New York, 1989.
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