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doi: 10.3934/ipi.2022020
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## A spectral target signature for thin surfaces with higher order jump conditions

 1 Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA 2 Department of Mathematical Sciences, University of Delaware, Newark, DE 19711, USA 3 Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA

In Memory of Professor Victor Isakov

Received  January 2022 Revised  March 2022 Early access April 2022

In this paper we consider the inverse problem of determining structural properties of a thin anisotropic and dissipative inhomogeneity in ${\mathbb R}^m$, $m = 2, 3$ from scattering data. In the asymptotic limit as the thickness goes to zero, the thin inhomogeneity is modeled by an open $m-1$ dimensional manifold (here referred to as screen), and the field inside is replaced by jump conditions on the total field involving a second order surface differential operator. We show that all the surface coefficients (possibly matrix valued and complex) are uniquely determined from far field patterns of the scattered fields due to infinitely many incident plane waves at a fixed frequency. Then we introduce a target signature characterized by a novel eigenvalue problem such that the eigenvalues can be determined from measured scattering data, adapting the approach in [20]. Changes in the measured eigenvalues are used to identified changes in the coefficients without making use of the governing equations that model the healthy screen. In our investigation the shape of the screen is known, since it represents the object being evaluated. We present some preliminary numerical results indicating the validity of our inversion approach

Citation: Fioralba Cakoni, Heejin Lee, Peter Monk, Yangwen Zhang. A spectral target signature for thin surfaces with higher order jump conditions. Inverse Problems and Imaging, doi: 10.3934/ipi.2022020
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##### References:
In the left column we show the scatterer $\Gamma$ (red curve) and the remainder of $\partial D$ as a green curve. Asterisks show the position of the random source points $z$ in $D$. In the right column we show the average $\ell_2$ norm of the regularized solution of the modified far field equation against the eigenparameter $\lambda$. The vertical lines mark the position of the true eigenvalues found by solving the interior eigenvalue problem. Top row: Dirichlet end condition. Bottom row: Neumann end condition
The layout of this figure is the same as in Fig. 2 except that the scatter is now the quarter circle shown in the left column. The same parameter values are used. Top row: Dirichlet end condition. Bottom row: Neumann end condition
Here we show the detection of eigenvalues for the half and quarter circle scatterers with Dirichlet end conditions and parameters given by $\mu = 0.2$, $\beta = 1$ and $\alpha = -0.2$. See Fig. 2 for a description of the symbols used. Top row: Half circle scatterer. Bottom: Quarter circle scatterer
Here we show the detection of eigenvalues for the half and quarter circle scatterers with Dirichlet end conditions and parameters given by $\mu = 0.2 = \beta = 2$ and $\alpha = -2$. The domain $D$ is now obtained by joining the end points of $\Gamma$ by a straight line. See Fig. 2 for a description of the symbols used. Top row: Half circle scatterer. Bottom: Quarter circle scatterer
An example of non-circular domains $D$ containing $\Gamma$. These domains are smoother than those in Fig. 5 and allow the approximation of more eigenvalues (the same parameters are used). Top row: Half circle scatterer. Bottom: Quarter circle scatterer
Changes in the first five eigenvalues (in magnitude) computed by the finite element eigenvalue solver for the half-circle scatterer and disk $D$ as functions of the parameters $\alpha$, $\beta$ and $\mu$
Predictions of the eigenvalues for the problem when $\alpha = -2$ and $\mu = 2$ and $\beta = 0.4, 0.5$ and 0.6. The shift in the eigenvalues predicted in Fig. 7 (middle graph) is evident in the large translation of the peaks for the three cases
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