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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  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 & Imaging, 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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] R. G. Newton, Variational principles for inverse scattering, Inverse Probl., 1 (1985), 371-380.  doi: 10.1088/0266-5611/1/4/008.  Google Scholar [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.  Google Scholar [8] M. Reed and B. Simon, Methods of Modern Mathematical Physics Ill: Scattering Theory, New York, San Francisco, London: Academic Press 1979.  Google Scholar [9] V. Serov, Fourier Series, Fourier Transform and Their Applications to Mathematical Physics, Springer, 2017.  Google Scholar [10] V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and Ill-Posed Problems Series, VSP, Utrecht, 1994. doi: 10.1515/9783110900095.  Google Scholar [11] P. Stefanov, Scattering and inverse scattering in ${\mathbf{R}}^n$. http://www.math.purdue.edu/ stefanov/publications/SCATTERING.pdf. Google Scholar [12] P. D. Stefanov, Inverse scattering problem for the wave equation with time dependent potential, C. R. Acad. Bulg. Sci., 40 (1987), 29-30.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] R. G. Newton, Variational principles for inverse scattering, Inverse Probl., 1 (1985), 371-380.  doi: 10.1088/0266-5611/1/4/008.  Google Scholar [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.  Google Scholar [8] M. Reed and B. Simon, Methods of Modern Mathematical Physics Ill: Scattering Theory, New York, San Francisco, London: Academic Press 1979.  Google Scholar [9] V. Serov, Fourier Series, Fourier Transform and Their Applications to Mathematical Physics, Springer, 2017.  Google Scholar [10] V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and Ill-Posed Problems Series, VSP, Utrecht, 1994. doi: 10.1515/9783110900095.  Google Scholar [11] P. Stefanov, Scattering and inverse scattering in ${\mathbf{R}}^n$. http://www.math.purdue.edu/ stefanov/publications/SCATTERING.pdf. Google Scholar [12] P. D. Stefanov, Inverse scattering problem for the wave equation with time dependent potential, C. R. Acad. Bulg. Sci., 40 (1987), 29-30.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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