We are concerned with the time harmonic elastic scattering in the presence of multiple small-scaled inclusions. The main property we use in this work is the local enhancement of scattering, which occurs at a specific incident frequency when the medium is perturbed with highly contrasted small inhomogeneities; for instance, one can consider contrast on the mass density. Such highly contrasting inclusions generate few local spots at their locations. These spots are generated as possible body waves related to elastic resonances. A family of these resonances is related to the eigenvalues of the elastic Newtonian operator.

\par Our goal is to derive the approximation of elastic scattered field for incident frequencies near to elastic resonances with suitable sufficient conditions. The dominating field generated due to the multiple interactions between a cluster of small inhomogeneities, of sub-wavelength size, is the Foldy-Lax field. The derived result has several applications, to mention a few, firstly in the theory of effective medium to design the materials with desired properties and, secondly, in elastic imaging to solve the inverse problem of recovering the properties of the background medium.

We consider Synthetic Aperture Radar (SAR) in which scattered waves, simultaneously emitted from a pair of stationary emitters, are measured along a flight track traversed by an aircraft. A linearized mathematical model of scattering is obtained using a Fourier integral operator. This model can then be used to form an image of the ground terrain using backprojection together with a carefully designed data acquisition geometry.

The data is composed of two parts, corresponding to the received signals from each emitter. A backprojection operator can be easily chosen that correctly reconstructs the singularities in the wave speed using just one emitter. One would expect this to lead to a reasonable image of the terrain. However, we expect that application of this backprojection operator to the data from the other emitter will lead to unwanted artifacts in the image. We analyse the operators associated with this situation, and use microlocal analysis to determine configurations of flight path and emitter locations so that we may mitigate the artifacts associated to such “cross talk” between the two emitters.

In this work we study an inverse problem for the Riemannian minimal surface equation, which is a quasilinear elliptic PDE. Consider a Riemannian manifold $(M,g)$ where $M=\mathbb{R}^n$ and the metric is a so called conformally transversally anisotropic metric i.e. $g=c(\hat{g}\oplus e)$, where $\hat{g}$ is a metric on $\mathbb{R}^{n-1}$. Let $u\colon\Omega\subset\mathbb{R}^{n-1}\to \mathbb{R}$ be a smooth function that satisfies the minimal surface equation. Assume that we can make boundary measurements on the graph of $u$, that is we know the Dirichlet-to-Neumann (DN) map which maps the boundary value $u|_{\partial\Omega}=f$ to the normal derivative $\partial_{\nu}u|_{\partial\Omega}=\hat{g}^{ij}\partial_{x_i}u\nu_j|_{\partial\Omega}$. The Dirichlet data $f$ is the height of minimal surface on the boundary. The normal derivative $\partial_{\nu}u|_{\partial\Omega}$ can be thought of as tension on the boundary caused by the minimal surface. In this talk we show that if we have knowledge of two DN maps corresponding to two different metrics in the same conformal class, then we can deduce that the metrics have the same Taylor series up to a constant multiplier.

This work connects some aspects of two previous articles, that is [1] and [2]. We use the technique of higher order linearization (see for example [3]) that has received increasing attention lately.

[1] J. Nurminen. An inverse problem for the minimal surface equation. Nonlinear Anal. 227,113163:19. 2023

[2] C. I. Cârstea, M. Lassas, T. Liimatainen, L. Oksanen. An inverse problems for the riemannian minimal surface equation, arXiv: 2203.09262:1–18. 2022

[3] M. Lassas, T. Liimatainen, Y.-H. Lin, M. Salo. Inverse problems for elliptic equations with power type nonlinearities. J. Math. Pures Appl. (9) 145: 44– 82. 2021