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Direct and inverse spectral problems for a star graph of Stieltjes strings damped at a pendant vertex
Inverse scattering and stability for the biharmonic operator
Department of Mathematics, Purdue University, West Lafayette, IN 47907 |
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}} $.
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. 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. |
[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. 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.
|
[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. 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. |
[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. 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.
|
[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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