The Calder\'on problem with partial data for conductivities with $3/2$ derivatives

TL;DR

This paper extends the uniqueness results in the Calderón problem to less regular conductivities with approximately 1.5 derivatives, showing that partial boundary data suffices for unique determination in dimensions three and higher.

Contribution

It demonstrates that conductivities with roughly 1.5 derivatives can be uniquely identified from partial boundary measurements, broadening the class of conductivities for which the Calderón problem is solvable.

Findings

Unique determination of conductivities with 3/2 derivatives from partial data

Extension of previous results to less regular conductivities

Applicability in dimensions n ≥ 3

Abstract

We extend a global uniqueness result for the Calder\'on problem with partial data, due to Kenig-Sj\"ostrand-Uhlmann, to the case of less regular conductivities. Specifically, we show that in dimensions $n\ge 3$, the knowledge of the Diricihlet-to-Neumann map, measured on possibly very small subsets of the boundary, determines uniquely a conductivity having essentially $3/2$ derivatives in an $L^2$ sense.