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Two-dimensional inverse scattering for quasi-linear biharmonic operator
1. | Biomimetics and Intelligent Systems Group, P.O. BOX 8000, FIN-90014 University of Oulu, Finland |
2. | Research Unit of Mathematical Sciences, P.O. BOX 3000, FIN-90014 University of Oulu, Finland |
3. | Department of Mathematics and Statistics, P.O. BOX 68, FI-00014 University of Helsinki |
The subject of this work concerns the classical direct and inverse scattering problems for quasi-linear perturbations of the two-dimensional biharmonic operator. The quasi-linear perturbations of the first and zero order might be complex-valued and singular. We show the existence of the scattering solutions to the direct scattering problem in the Sobolev space $ W^1_{\infty}( \mathbb{{R}}^2) $. Then the inverse scattering problem can be formulated as follows: does the knowledge of the far field pattern uniquely determine the unknown coefficients for given differential operator? It turns out that the answer to this classical question is affirmative for quasi-linear perturbations of the biharmonic operator. Moreover, we present a numerical method for the reconstruction of unknown coefficients, which from the practical point of view can be thought of as recovery of the coefficients from fixed energy measurements.
References:
[1] |
S. Agmon,
Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2 (1975), 151-218.
|
[2] |
T. M. Buzug, Computed Tomography: From Photon Statistics to Modern Cone-Beam CT, Springer, Berlin Heidelberg, 2008. Google Scholar |
[3] |
F. Cakoni and D. Colton, A Qualitative Approach in Inverse Scattering Theory, Springer, New York, 2014.
doi: 10.1007/978-1-4614-8827-9. |
[4] |
G. Fotopoulos and M. Harju,
Inverse scattering with fixed observation angle data in 2D, Inv. Prob. Sci. Eng., 25 (2017), 1492-1507.
doi: 10.1080/17415977.2016.1267170. |
[5] |
G. Fotopoulos, M. Harju and V. Serov,
Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D, Inverse Problems ans Imaging, 7 (2013), 183-197.
doi: 10.3934/ipi.2013.7.183. |
[6] |
G. Fotopoulos and V. Serov,
Inverse fixed energy scattering problem for the two-dimensional nonlinear Schrödinger operator, Inv. Prob. Sci. Eng., 24 (2016), 692-710.
doi: 10.1080/17415977.2015.1055263. |
[7] |
F. Gazzola, H.-Ch. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer-Verlag, Berlin Heidelberg, 2010.
doi: 10.1007/978-3-642-12245-3. |
[8] |
L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, New Jersey, 2004. |
[9] |
M. Harju, On the Direct and Inverse Scattering Problems for a Nonlinear Three-Dimensional Schrödinger Equation, PhD-thesis, University of Oulu, 2010. Google Scholar |
[10] |
K. Krupchyk, M. Lassas and G. Uhlmann,
Determining a first order perturbation of the biharmonic operator by partial boundary measurements, Journal of Functional Analysis, 262 (2012), 1781-1801.
doi: 10.1016/j.jfa.2011.11.021. |
[11] |
K. Krupchyk, M. Lassas and G. Uhlmann,
Inverse boundary value problems for the perturbed polyharmonic operator, Trans Amer. Math. Soc., 366 (2014), 95-112.
doi: 10.1090/S0002-9947-2013-05713-3. |
[12] |
N. N. Lebedev, Special Functions and Their Applications, Dover Publications, 1972. |
[13] |
B. Pausander, Scattering for the beam equation in low dimensions, Indiana Univ. Math. J., 59 (2010), 791–822. arXiv: 0903.3777v2 [math.AP].
doi: 10.1512/iumj.2010.59.3966. |
[14] |
V. S. Serov, An inverse Born approximation for the general nonlinear Schrödinger operator on the line, Journal of Physics A: Mathematical and Theoretical, 42 (2009), 332002.
doi: 10.1088/1751-8113/42/33/332002. |
[15] |
V. Serov, M. Harju and G. Fotopoulos, Direct and inverse scattering for nonlinear Schrödinger equation in 2D, Journal of Mathematical Physics, 53 (2012), 123522.
doi: 10.1063/1.4769825. |
[16] |
T. Tyni, Numerical results for Saito's uniqueness theorem in inverse scattering theory, Inverse Problems, 35 (2020), 065002.
doi: 10.1088/1361-6420/ab7d2d. |
[17] |
T. Tyni and V. Serov,
Scattering problems for perturbations of the multidimensional biharmonic operator, Inverse Problems and Imaging, 12 (2018), 205-227.
doi: 10.3934/ipi.2018008. |
[18] |
T. Tyni and M. Harju, Inverse backscattering problem for perturbations of biharmonic operator, Inverse Problems, 33 (2017), 105002.
doi: 10.1088/1361-6420/aa873e. |
[19] |
T. Tyni and V. Serov,
Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line, Inverse Problems and Imaging, 13 (2019), 159-175.
doi: 10.3934/ipi.2019009. |
[20] |
E. Zeidler, Applied Functional Analysis, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-0821-1. |
show all references
References:
[1] |
S. Agmon,
Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2 (1975), 151-218.
|
[2] |
T. M. Buzug, Computed Tomography: From Photon Statistics to Modern Cone-Beam CT, Springer, Berlin Heidelberg, 2008. Google Scholar |
[3] |
F. Cakoni and D. Colton, A Qualitative Approach in Inverse Scattering Theory, Springer, New York, 2014.
doi: 10.1007/978-1-4614-8827-9. |
[4] |
G. Fotopoulos and M. Harju,
Inverse scattering with fixed observation angle data in 2D, Inv. Prob. Sci. Eng., 25 (2017), 1492-1507.
doi: 10.1080/17415977.2016.1267170. |
[5] |
G. Fotopoulos, M. Harju and V. Serov,
Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D, Inverse Problems ans Imaging, 7 (2013), 183-197.
doi: 10.3934/ipi.2013.7.183. |
[6] |
G. Fotopoulos and V. Serov,
Inverse fixed energy scattering problem for the two-dimensional nonlinear Schrödinger operator, Inv. Prob. Sci. Eng., 24 (2016), 692-710.
doi: 10.1080/17415977.2015.1055263. |
[7] |
F. Gazzola, H.-Ch. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer-Verlag, Berlin Heidelberg, 2010.
doi: 10.1007/978-3-642-12245-3. |
[8] |
L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, New Jersey, 2004. |
[9] |
M. Harju, On the Direct and Inverse Scattering Problems for a Nonlinear Three-Dimensional Schrödinger Equation, PhD-thesis, University of Oulu, 2010. Google Scholar |
[10] |
K. Krupchyk, M. Lassas and G. Uhlmann,
Determining a first order perturbation of the biharmonic operator by partial boundary measurements, Journal of Functional Analysis, 262 (2012), 1781-1801.
doi: 10.1016/j.jfa.2011.11.021. |
[11] |
K. Krupchyk, M. Lassas and G. Uhlmann,
Inverse boundary value problems for the perturbed polyharmonic operator, Trans Amer. Math. Soc., 366 (2014), 95-112.
doi: 10.1090/S0002-9947-2013-05713-3. |
[12] |
N. N. Lebedev, Special Functions and Their Applications, Dover Publications, 1972. |
[13] |
B. Pausander, Scattering for the beam equation in low dimensions, Indiana Univ. Math. J., 59 (2010), 791–822. arXiv: 0903.3777v2 [math.AP].
doi: 10.1512/iumj.2010.59.3966. |
[14] |
V. S. Serov, An inverse Born approximation for the general nonlinear Schrödinger operator on the line, Journal of Physics A: Mathematical and Theoretical, 42 (2009), 332002.
doi: 10.1088/1751-8113/42/33/332002. |
[15] |
V. Serov, M. Harju and G. Fotopoulos, Direct and inverse scattering for nonlinear Schrödinger equation in 2D, Journal of Mathematical Physics, 53 (2012), 123522.
doi: 10.1063/1.4769825. |
[16] |
T. Tyni, Numerical results for Saito's uniqueness theorem in inverse scattering theory, Inverse Problems, 35 (2020), 065002.
doi: 10.1088/1361-6420/ab7d2d. |
[17] |
T. Tyni and V. Serov,
Scattering problems for perturbations of the multidimensional biharmonic operator, Inverse Problems and Imaging, 12 (2018), 205-227.
doi: 10.3934/ipi.2018008. |
[18] |
T. Tyni and M. Harju, Inverse backscattering problem for perturbations of biharmonic operator, Inverse Problems, 33 (2017), 105002.
doi: 10.1088/1361-6420/aa873e. |
[19] |
T. Tyni and V. Serov,
Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line, Inverse Problems and Imaging, 13 (2019), 159-175.
doi: 10.3934/ipi.2019009. |
[20] |
E. Zeidler, Applied Functional Analysis, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-0821-1. |






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