Inverse Problems for Nonlinear Elliptic Partial Differential Equations and Fractional Anisotropic Calderon Problem

Abstract

In the first part of the talk, the speaker will discuss some recent progress for inverse boundary problems for nonlinear elliptic partial differential equations. The focus will be on inverse problems for isotropic quasilinear conductivity equations, as well as nonlinear Schrodinger and magnetic Schrodinger equations. In particular, she and her research collaborators see that the presence of a nonlinearity may actually help, allowing one to solve inverse problems in situations where the corresponding linear counterpart is open. In the second part of the talk, the speaker shall discuss the fractional anisotropic Calderon problem on closed Riemannian manifolds of dimensions two and higher. Specifically, they show that the knowledge of the local source-to-solution map for the fractional Laplacian, given on an arbitrary small open nonempty a priori known subset of a smooth closed Riemannian manifold, determines the Riemannian manifold up to an isometry. This can be viewed as a nonlocal analog of the anisotropic Calderon problem in the setting of closed Riemannian manifolds, which is wide open in dimensions three and higher. This talk is based on joint works with Catalin CARSTEA, Ali FEIZMOHAMMADI, Tuhin GHOSH, Yavar KIAN, and Gunther UHLMANN.

About the Speaker

Prof. Katya KRUPCHYK received her PhD in Mathematics from the Belarusian State University in 2003. She continued her research as a Postdoctoral Fellow at the University of Eastern Finland during 2004-2005, supported by the Centre for International Mobility grant. In 2005, she was awarded a competitive 3-year Postdoctoral Position and Fellowship from the Academy of Finland and in 2008, she was awarded a competitive 5-year Academy of Finland Research Fellow Position and Fellowship. During 2008-2014, she worked at the University of Helsinki as an Academy of Finland Research Fellow and a University Researcher. In 2013, she was awarded the title of Docent from the University of Helsinki. In 2014, she joined the University of California, Irvine and is currently a Professor of Mathematics there.

Prof. Krupchyk’s research interests include partial differential equations, inverse problems, microlocal analysis, and spectral theory. She has served on the editorial boards of Inverse Problems and Imaging and SIAM Journal on Mathematical Analysis. She was awarded the Research Professorship by the Mathematical Sciences Research Institute in 2019.

For Attendees' Attention

This talk will be held online via Zoom. To attend, please join the Zoom meeting at https://hkust.zoom.us/j/95839398543 (Meeting ID: 958 3939 8543 / Passcode: iasip2022).

About the Program

For more information, please refer to the program website at https://iasprogram.hkust.edu.hk/inverseproblems/.