Research Projects

The project focuses on addressing two high-frequency inverse problems involving the Laplace-Beltrami operator, which governs wave propagation on a Riemannian manifold and one problem related to elastic wave equation, which leads to study integral geometric problems on Finsler setting. The first part of the project aims to recover the scattering relation from the Dirichlet-to-Neumann (DN) map associated with the high-frequency Laplace-Beltrami operator. The equation under consideration is Laplace Beltrami at high frequency on a compact Riemannian manifolds with boundary. The scattering relation describes how an incoming point and direction on the boundary of the manifold are mapped to an outgoing point and direction after interacting with the interior of the manifold. This relation provides essential information about the geometry and structure of the manifold's interior, and the goal is to deduce it using only boundary measurements encapsulated in the DN map. The second part of the project considers a more complex high-frequency inverse problem for a fourth-order Laplace-Beltrami operator. The equation in this case is represents the fourth-order Laplace-Beltrami operator with a first order and zeroth order perturbation The goal here is to recover a first order and zeroth order unknown efficients from the DN map at a high fixed frequency. These problems are challenging because they involve reconstructing detailed information about the manifold's geometry and internal features from indirect and limited boundary data. The solutions require advanced mathematical tools, including microlocal analysis and techniques for high-frequency asymptotics. The results of this project have potential applications in mathematical physics and imaging, where understanding the internal properties of a medium from boundary observations is a key problem. On the third problem, the goal of the study is to recover the largest wave speed of solutions to this equation, which corresponds to the maximum velocity at which elastic waves propagate through the medium. The wave speeds are determined by the eigenvalues of the Christoffel matrix, derived from the elastic tensor, which is fourth order anisotropic tensor. Specifically, the largest wave speed is associated with the fastest propagating mode, typically the P-wave (primary or compressional wave). We wish to implement propagation singularities to recover the scattering relation (possible polarized one) related to a largest wave speed.