Outreach

My research focuses on the analysis of partial differential equations (PDEs) and the inverse problems associated with them. A central theme of my work is to recover hidden structures in PDE models from indirect measurements, a challenge that naturally arises in applications such as medical imaging, geophysics, and biology. I am particularly interested in PDEs with nonlocal or nonlinear features, and more recently I have begun studying problems involving non-uniformly elliptic equations. In many of these inverse problems, the data are encoded in operators such as the Dirichlet-to-Neumann map or the source-to-solution map.

More concretely, my research covers the following directions:

• Nonlocal generalizations of the Calderón problem (e.g., fractional conductivity equation)

• Interaction of nonlinear and nonlocal effects in inverse problems (e.g., fractional p-Laplacian, fractional p-biharmonic operators, nonlocal porous medium equation)

• Nonlocal approaches to local inverse problems (e.g., Schrödinger equation in transversally anisotropic geometries)

• Inverse problems for evolution equations (e.g., local and nonlocal diffusion/waves, nonlinear Schrödinger systems)

• Stability estimates in inverse problems (e.g., relativistic wave equations, fractional conductivity equation)

• Unique continuation and Runge approximation properties (e.g., unique continuation of the fractional Laplacian in Bessel potential spaces)

• Inverse problems for non-uniformly elliptic PDEs (e.g., double phase problem)