Title: Calderón Type Inverse Problems for Integro-differential Operators

Teacher: li LI 李力（Tsinghua University）

Time: 16th, Jun.-18th, Jun (19:30-21:30)

Classroom: 2nd Teaching Building, Room 206 

Abstract：This minicourse introduces the mathematical framework, proof strategies, and modern developments of inverse problems for nonlocal equations, tracking their evolution from classical elliptic equations to contemporary space-time fractional equations. The curriculum is structured into three consecutive lectures:

1. From Classical to Fractional Calderón Problems

We review the foundational classical Calderón electrical impedance problem, exploring Complex Geometrical Optics (CGO) solutions and partial data unique determination via Carleman estimates. We then transition to the fractional analogue, detailing how the nonlocality of the fractional Laplacian shifts the measurement paradigm to the exterior domain. The session culminates in proving the Unique Continuation Property (UCP) and establishing the associated Runge approximation property via duality arguments.

2. Variable Coefficient Recoveries under Local Perturbations

This session explores generalizations of the fractional Calderón problem. We analyze local perturbations driven by linear partial differential operators, presenting methods to achieve the unique determination of variable coefficients from exterior Dirichlet-to-Neumann (DN) maps. We also specifically target the semilinear variant of the fractional Calderón problem governed by power-type nonlinearities， and associated linearization techniques will be covered.

3. Evolutionary Settings and Space-Time Fractional Dynamics

The final block shifts focus to time-dependent regimes and evolutionary variants. We formalize typical inverse problems for the memory-dependent process  modeled by Caputo fractional derivatives. Finally, we highlight recent developments in formulating and solving inverse problems for both coupled and uncoupled space-time fractional operators.