Stability of an inverse source problem for the damped biharmonic plate equation

This paper is concerned with the stability of the inverse source problem for the damped biharmonic plate equation in three dimensions. The stability estimate consists of the Lipschitz type data discrepancy and the high frequency tail of the source function, where the latter decreases as the upper bound of the frequency increases. The stability also shows exponential dependence on the damping coefficient. The ingredients of the analysis include Carleman estimates and time decay estimates for the damped plate wave equation to obtain an exact observability bound, and the study of the resonance-free region and an upper bound of the resolvent for the biharmonic operator with respect to the complex wavenumber.

[1]  Jin Cheng,et al.  Increasing stability in the inverse source problem with many frequencies , 2016 .

[2]  On increasing stability in the two dimensional inverse source scattering problem with many frequencies , 2017, 1712.08696.

[3]  F. Gazzola,et al.  Polyharmonic Boundary Value Problems , 2010 .

[4]  Gang Bao,et al.  Stability for the inverse source problems in elastic and electromagnetic waves , 2017, Journal de Mathématiques Pures et Appliquées.

[5]  Junshan Lin,et al.  A multi-frequency inverse source problem , 2010 .

[6]  Masahiro Yamamoto,et al.  Lipschitz stability in inverse problems for a Kirchhoff plate equation , 2007, Asymptot. Anal..

[7]  J. Rousseau,et al.  Spectral inequality and resolvent estimate for the bi-Laplace operator , 2015, Journal of the European Mathematical Society.

[8]  Peijun Li,et al.  Increasing stability for the inverse source scattering problem with multi-frequencies , 2016, 1607.06953.

[9]  Victor Isakov,et al.  Increasing Stability in Acoustic and Elastic Inverse Source Problems , 2018, SIAM J. Math. Anal..

[10]  V. Serov,et al.  Scattering problems for perturbations of the multidimensional biharmonic operator , 2018 .

[11]  G. Uhlmann,et al.  Inverse boundary value problems for the perturbed polyharmonic operator , 2011, 1102.5542.

[12]  Ross C. McPhedran,et al.  Wave scattering by platonic grating stacks , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[13]  Peijun Li,et al.  Stability for the Acoustic Inverse Source Problem in Inhomogeneous Media , 2020, SIAM J. Appl. Math..

[14]  G. Uhlmann,et al.  Determining a first order perturbation of the biharmonic operator by partial boundary measurements , 2011, 1103.0113.

[15]  Victor Isakov,et al.  Increasing Stability in the Inverse Source Problem with Attenuation and Many Frequencies , 2018, SIAM J. Appl. Math..

[16]  Time evolution of the scattering data for a fourth-order linear differential operator , 2008, 0805.3554.

[17]  Sharp stability estimates of harmonic continuation along lines , 2000 .

[18]  Peijun Li,et al.  Stability for an Inverse Source Problem of the Biharmonic Operator , 2021, SIAM J. Appl. Math..