# American Institute of Mathematical Sciences

April  2021, 15(2): 271-283. doi: 10.3934/ipi.2020064

## Inverse scattering and stability for the biharmonic operator

 Department of Mathematics, Purdue University, West Lafayette, IN 47907

Received  April 2020 Revised  August 2020 Published  April 2021 Early access  November 2020

Fund Project: The author is partially supported by a NSF DMS grants No. 1600327 and 1900475

We investigate the inverse scattering problem of the perturbed biharmonic operator by studying the recovery process of the magnetic field ${\mathbf{A}}$ and the potential field $V$. We show that the high-frequency asymptotic of the scattering amplitude of the biharmonic operator uniquely determines ${\rm{curl}}\ {\mathbf{A}}$ and $V-\frac{1}{2}\nabla\cdot{\mathbf{A}}$. We study the near-field scattering problem and show that the high-frequency asymptotic expansion up to an error $\mathcal{O}(\lambda^{-4})$ recovers above two quantities with no additional information about ${\mathbf{A}}$ and $V$. We also establish stability estimates for ${\rm{curl}}\ {\mathbf{A}}$ and $V-\frac{1}{2}\nabla\cdot{\mathbf{A}}$.

Citation: Siamak RabieniaHaratbar. Inverse scattering and stability for the biharmonic operator. Inverse Problems and Imaging, 2021, 15 (2) : 271-283. doi: 10.3934/ipi.2020064
##### References:
 [1] F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer-Verlag Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-12245-3. [2] G. M. Henkin and R. G. Novikov, The $\bar{\partial}$-Equation in the multi-dimensional inverse scattering problem, Usp. Mat. Nauk., 42 (1987), 93-152. [3] A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Monographs and Surveys in Pure and Applied Mathematics, 123, Chapman Hall/CRC-press, 2001. doi: 10.1201/9781420036220. [4] K. Krupchyk, M. Lassas and G. Uhlmann, Determining a first order perturbation of the biharmonic operator by partial boundary measurements, J. Funct. Anal., 262 (2012), 1781–1801. doi: 10.1016/j.jfa.2011.11.021. [5] G. Nakamura, Z. Uhlmann and G. Sun, Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field, Math. Ann., 303 (1995), 377-388.  doi: 10.1007/BF01460996. [6] R. G. Newton, Variational principles for inverse scattering, Inverse Probl., 1 (1985), 371-380.  doi: 10.1088/0266-5611/1/4/008. [7] B. Pausader, Scattering for the defocusing beam equation in low dimensions, Indiana Univ. Math. J., 59 (2010), 791-822.  doi: 10.1512/iumj.2010.59.3966. [8] M. Reed and B. Simon, Methods of Modern Mathematical Physics Ill: Scattering Theory, New York, San Francisco, London: Academic Press 1979. [9] V. Serov, Fourier Series, Fourier Transform and Their Applications to Mathematical Physics, Springer, 2017. [10] V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and Ill-Posed Problems Series, VSP, Utrecht, 1994. doi: 10.1515/9783110900095. [11] P. Stefanov, Scattering and inverse scattering in ${\mathbf{R}}^n$. http://www.math.purdue.edu/ stefanov/publications/SCATTERING.pdf. [12] P. D. Stefanov, Inverse scattering problem for the wave equation with time dependent potential, C. R. Acad. Bulg. Sci., 40 (1987), 29-30. [13] P. D. Stefanov, Uniqueness of the multi-dimensional inverse scattering problem for time dependent potentials, Math. Z., 201 (1989), 541-559.  doi: 10.1007/BF01215158. [14] Z. Q. Sun, An inverse boundary value problem for Schrödinger operators with vector potentials, Trans. Amer. Math. Soc., 338 (1993), 953-969.  doi: 10.2307/2154438. [15] T. Tyni and V. Serov, Scattering problems for perturbations of the multidimensional biharmonic operator, Inverse Problem and Imaging, 12 (2018), 205-227.  doi: 10.3934/ipi.2018008. [16] T. Tyni and M. Harju, Inverse backscattering problem for perturbations of biharmonic operator, Inverse Problems, 33 (2017), 105002, 20 pp. doi: 10.1088/1361-6420/aa873e.

show all references

##### References:
 [1] F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer-Verlag Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-12245-3. [2] G. M. Henkin and R. G. Novikov, The $\bar{\partial}$-Equation in the multi-dimensional inverse scattering problem, Usp. Mat. Nauk., 42 (1987), 93-152. [3] A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Monographs and Surveys in Pure and Applied Mathematics, 123, Chapman Hall/CRC-press, 2001. doi: 10.1201/9781420036220. [4] K. Krupchyk, M. Lassas and G. Uhlmann, Determining a first order perturbation of the biharmonic operator by partial boundary measurements, J. Funct. Anal., 262 (2012), 1781–1801. doi: 10.1016/j.jfa.2011.11.021. [5] G. Nakamura, Z. Uhlmann and G. Sun, Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field, Math. Ann., 303 (1995), 377-388.  doi: 10.1007/BF01460996. [6] R. G. Newton, Variational principles for inverse scattering, Inverse Probl., 1 (1985), 371-380.  doi: 10.1088/0266-5611/1/4/008. [7] B. Pausader, Scattering for the defocusing beam equation in low dimensions, Indiana Univ. Math. J., 59 (2010), 791-822.  doi: 10.1512/iumj.2010.59.3966. [8] M. Reed and B. Simon, Methods of Modern Mathematical Physics Ill: Scattering Theory, New York, San Francisco, London: Academic Press 1979. [9] V. Serov, Fourier Series, Fourier Transform and Their Applications to Mathematical Physics, Springer, 2017. [10] V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and Ill-Posed Problems Series, VSP, Utrecht, 1994. doi: 10.1515/9783110900095. [11] P. Stefanov, Scattering and inverse scattering in ${\mathbf{R}}^n$. http://www.math.purdue.edu/ stefanov/publications/SCATTERING.pdf. [12] P. D. Stefanov, Inverse scattering problem for the wave equation with time dependent potential, C. R. Acad. Bulg. Sci., 40 (1987), 29-30. [13] P. D. Stefanov, Uniqueness of the multi-dimensional inverse scattering problem for time dependent potentials, Math. Z., 201 (1989), 541-559.  doi: 10.1007/BF01215158. [14] Z. Q. Sun, An inverse boundary value problem for Schrödinger operators with vector potentials, Trans. Amer. Math. Soc., 338 (1993), 953-969.  doi: 10.2307/2154438. [15] T. Tyni and V. Serov, Scattering problems for perturbations of the multidimensional biharmonic operator, Inverse Problem and Imaging, 12 (2018), 205-227.  doi: 10.3934/ipi.2018008. [16] T. Tyni and M. Harju, Inverse backscattering problem for perturbations of biharmonic operator, Inverse Problems, 33 (2017), 105002, 20 pp. doi: 10.1088/1361-6420/aa873e.
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