Revisiting the Anisotropic Fractional Calder\'on Problem Using the Caffarelli-Silvestre Extension

TL;DR

This paper explores the anisotropic fractional Calderón problem using the Caffarelli-Silvestre extension, establishing equivalences between heat, wave, and Poisson kernels, and connecting local and nonlocal inverse problems.

Contribution

It provides an alternative proof for kernel recovery via the Caffarelli-Silvestre approach and links different perspectives in the anisotropic fractional Calderón problem.

Findings

Heat and wave kernels can be recovered from each other.

Poisson kernel offers a direct approach bypassing heat kernel recovery.

Connections between local and nonlocal Calderón problems are established.

Abstract

We revisit the source-to-solution anisotropic fractional Calder\'on problem introduced and analyzed in [FGKU21] and [F21]. Using the Caffarelli-Silvestre interpretation of the fractional Laplacian, we provide an alternative argument for the recovery of the heat and wave kernels from [FGKU21]. This shows that in the setting of the source-to-solution anisotropic fractional Calder\'on problem the heat and Caffarelli-Silvestre approach give rise to equivalent perspectives and that each kernel can be recovered from the other. Moreover, we also discuss the Dirichlet-to-Neumann anisotropic source-to-solution problem and provide a direct link between the Dirichlet Poisson kernel and the wave kernel. This illustrates that it is also possible to argue completely on the level of the Poisson kernel, bypassing the recovery of the heat kernel as an additional auxiliary step. Last but not least, as in [CGRU23], we relate the local and nonlocal source-to-solution Calder\'on problems.