Fractional Calder\'on problems and Poincar\'e inequalities on unbounded domains

TL;DR

This paper extends uniqueness results for the fractional Calderón problem to unbounded domains, characterizes when partial data inverse problems are unique, and establishes Poincaré inequalities and approximation results for fractional Laplacians.

Contribution

It generalizes the fractional Calderón problem to all domains with nonempty exterior and develops new theoretical tools for inverse problems involving fractional operators.

Findings

Uniqueness results for fractional Calderón problems on unbounded domains.

Poincaré inequalities for fractional Laplacians of any order.

Runge approximation and exterior determination results.

Abstract

We generalize many recent uniqueness results on the fractional Calder\'on problem to cover the cases of all domains with nonempty exterior. The highlight of our work is the characterization of uniqueness and nonuniqueness of partial data inverse problems for the fractional conductivity equation on domains that are bounded in one direction for conductivities supported in the whole Euclidean space and decaying to a constant background conductivity at infinity. We generalize the uniqueness proof for the fractional Calder\'on problem by Ghosh, Salo and Uhlmann to a general abstract setting in order to use the full strength of their argument. This allows us to observe that there are also uniqueness results for many inverse problems for higher order local perturbations of a lower order fractional Laplacian. We give concrete example models to illustrate these curious situations and prove Poincar\'e inequalities for the fractional Laplacians of any order on domains that are bounded in one direction. We establish Runge approximation results in these general settings, improve regularity assumptions also in the cases of bounded sets and prove general exterior determination results. Counterexamples to uniqueness in the inverse fractional conductivity problem with partial data are constructed in another companion work.