On some partial data Calder\'on type problems with mixed boundary conditions

TL;DR

This paper investigates the simultaneous recovery of bulk and boundary potentials in degenerate elliptic equations, connecting local and nonlocal Calderón problems, and establishes uniqueness results using Runge approximation and CGO solutions.

Contribution

It introduces new Runge approximation and CGO solutions for degenerate elliptic equations, advancing the understanding of Calderón type inverse problems with mixed boundary conditions.

Findings

Proved simultaneous bulk and boundary Runge approximation results.

Established uniqueness for localized bulk and boundary potentials.

Constructed CGO solutions via a novel Carleman estimate.

Abstract

In this article we consider the simultaneous recovery of bulk and boundary potentials in (degenerate) elliptic equations modelling (degenerate) conducting media with inaccessible boundaries. This connects local and nonlocal Calder\'on type problems. We prove two main results on these type of problems: On the one hand, we derive simultaneous bulk and boundary Runge approximation results. Building on these, we deduce uniqueness for localized bulk and boundary potentials. On the other hand, we construct a family of CGO solutions associated with the corresponding equations. These allow us to deduce uniqueness results for arbitrary bounded, not necessarily localized bulk and boundary potentials. The CGO solutions are constructed by duality to a new Carleman estimate.