A Reduction of the Fractional Calder\'on Problem to the Local Calder\'on Problem by Means of the Caffarelli-Silvestre Extension

Abstract

We relate the (anisotropic) variable coefficient local and nonlocal Calder\'on problems by means of the Caffarelli-Silvestre extension. In particular, we prove that (partial) Dirichlet-to-Neumann data for the fractional Calder\'on problem in three and higher dimensions determine the (full) Dirichlet-to-Neumann data for the local Calder\'on problem. As a consequence, any (variable coefficient) uniqueness result for the local problem also implies a uniqueness result for the nonlocal problem. Moreover, our approach is constructive and associated Tikhonov regularization schemes can be used to recover the data. Finally, we highlight obstructions for reversing this procedure, which essentially consist of two one-dimensional averaging processes.