Inverse problems for fractional Laplacian with non-local lower order perturbation

Abstract:  In this talk I will be presenting a recent work, jointly done with Prof. Gunther Uhlmann and Dr. Tuhin Ghosh, where we consider a non-local inverse problem and determine more than one lower order coefficients from the associate Cauchy data. We consider the following perturbed fractional Laplacian operator  

                                             \((-\Delta)^t + (-\Delta)_{\Omega}^{{s}/{2}} \ b (-\Delta)_{\Omega}^{{s}/{2}} + c, \quad 0<s<t<1\)

on a bounded Lipschitz domain \(\Omega \subset \mathbb{R}^n\). Apart from the global non-locality in the principal part, our operator exhibits regional non-locality in its lower order regional fractional Laplacian perturbation term \((-\Delta)^{s/2}_{\Omega}\).

We find that by knowing the corresponding Dirichlet to Neumann map (D-N map) on the exterior domain \(\mathbb{R}^n \setminus \Omega\), it is possible to determine the lower order coefficients \(`b'\) ,\(`c'\) in \(\Omega\). We also discuss the recovery of \(`b'\), `\(c\)' from a single measurement and its limitations