We consider the inverse problem of recovering an unknown, spatially-dependent coefficient $q(x)$ from the fractional order equation $\mathbb{L}_\alpha u = f$ defined in a region of $\mathbb{R}^2$ from boundary information. Here $\mathbb{L_\alpha} ={D}^{\alpha_x}_x +{D}^{\alpha_y}_y +q(x)$ where the operators ${D}^{\alpha_x}_x$, ${D}^{\alpha_y}_y$ denote fractional derivative operators based on the Abel fractional integral. In the classical case this reduces to $-\triangle u + q(x)u = f$ and this has been a well-studied problem. We develop both uniqueness and reconstruction results and show how the ill-conditioning of this inverse problem depends on the geometry of the region and the fractional powers $\alpha_x$ and $\alpha_y$.

We give several examples of solutions inverse problems involving lon-range interactions whosecorresponding local problem is not solved.

The main purpose of this talk is to discuss two different nonlocal variants of the $p$-Calderón problem.

In the first model the nonlocal operator under consideration is a weighted fractional $p$-Laplacian and, similarly as for the $p$-Laplacian in dimensions $n\geq 3$, it is an open problem, whether it satisfies a unique continuation principle (UCP). However, it will be explained that the variational structure of the problem is still sufficiently nice that one can explicitly reconstruct the weight $\sigma(x,y)$ on the diagonal $D=\{(x,x): x\in W\}$ of the measurement set $W$. This reconstruction formula establishes a global uniqueness result for separable, real analytic coefficients [1].

In the second model, we consider the (anisotropic) fractional $p$-biharmonic operator, which naturally appears in the variational characterization of the optimal fractional Poincaré constant in Bessel potential spaces $H^{s,p}$. In contrast to the one above, this operator satisfies the UCP and so heuristically corresponds to the $p$-Laplacian in dimension $n=2$. Finally, we explain how this can be used to establish a global uniqueness result of the related inverse problem under a monotonicity condition [2].

[1] M. Kar, Y. Lin, and P. Zimmermann. Determining coefficients for a fractional $ p $-Laplace equation from exterior measurements, arXiv:2212.03057, 2022.

[2] M. Kar, J. Railo, and P. Zimmermann. The fractional $ p\, $-biharmonic systems: optimal Poincaré constants, unique continuation and inverse problems, arXiv:2208.09528, 2022.

We study an inverse problem for the fractional anisotropic conductivity equation. Our nonlocal operator is based on the well-developed theory of nonlocal vector calculus, and differs substantially from other generalizations of the classical anisotropic conductivity operator obtained spectrally. We show that the anisotropic conductivity matrix can be recovered uniquely from fractional Dirichlet-to-Neumann data up to a natural gauge. Our analysis makes use of techniques recently developed for the study of the isotropic fractional elasticity equation, and generalizes them to the case of non-separable, anisotropic conductivities. The motivation for our study stems from its relation to the classical anisotropic Calderòn problem, which is one of the main open problems in the field.