Fractional Calderón Problem

• The fractional Calderón problem is an inverse boundary value problem that determines unknown interior parameters using fractional powers of divergence-form operators.
• It employs advanced analytic techniques, such as the Caffarelli–Silvestre extension, to recover variable conductivities and potentials from nonlocal Dirichlet-to-Neumann data.
• Recent advances reveal its deep connections with the classical Calderón problem, highlighting limitations and stability challenges in higher-dimensional and anisotropic contexts.

The fractional Calderón problem is a class of inverse boundary value problems involving nonlocal elliptic PDEs based on fractional powers of divergence-form operators with variable or anisotropic coefficients. The central objective is to determine unknown interior parameters (such as conductivities or potentials) from prescribed nonlocal Dirichlet-to-Neumann data measured on disjoint open subsets of the domain's exterior. Recent advances have shown the profound connections between the fractional and classical (local) Calderón problems—especially via analytic tools such as the Caffarelli–Silvestre extension. These yields reduction results, constructive recovery methods, and fundamental limitations on the invertibility and stability of nonlocal operators in multidimensional settings.

1. Fractional vs. Classical Calderón Problems

Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain with $$\Omega_e = \mathbb{R}^n \setminus \overline{\Omega}$$ an open exterior. The classical Calderón problem for variable conductivity seeks to determine an unknown symmetric, positive definite matrix-valued function $\sigma(x):\Omega\to\mathbb{R}^{n\times n}$ from boundary Dirichlet-to-Neumann measurements: $$\begin{cases} \nabla \cdot \sigma \nabla v = 0 & \text{in }\Omega \ v = g & \text{on }\partial\Omega \end{cases}$$ with map [ \Lambda_\sigma: H^{1/2}(\partial\Omega) \rightarrow H^{-1/2}(\partial\Omega), \quad g \mapsto \sigma \nabla v \