A partial data result for less regular conductivities in admissible geometries

TL;DR

This paper advances the Calderón problem by demonstrating that partial boundary measurements on specific geometries can determine conductivities with fractional derivatives, strengthening previous results.

Contribution

It provides a new partial data uniqueness result for conductivities with fractional regularity in admissible geometries, extending prior work by Kenig, Sjöstrand, and Uhlmann.

Findings

Boundary measurements determine conductivities with 3/2 derivatives.

Strengthens existing partial data uniqueness results.

Applicable to conformally embedded product geometries.

Abstract

We consider the Calder\'on problem with partial data in certain admissible geometries, that is, on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. We show that measuring the Dirichlet-to-Neumann map on roughly half of the boundary determines a conductivity that has essentially 3/2 derivatives. As a corollary, we strengthen a partial data result due to Kenig, Sj\"ostrand, and Uhlmann.