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.