The Calder\'on Problem for Quasilinear Conductivities of Conformally Transversally Anisotropic Media

TL;DR

This paper extends Calderón's inverse problem to quasilinear conductivities on anisotropic manifolds, showing unique determination of the conductivity and its derivatives from boundary measurements, especially when the conductivity is analytic.

Contribution

It proves the unique recoverability of quasilinear conductivities on conformally transversally anisotropic manifolds from boundary data, including derivatives and the entire conductivity if analytic.

Findings

Derivatives of the conductivity are uniquely determined by boundary measurements.

The entire conductivity function can be recovered if it is analytic in the electric field variable.

The results apply to conductivities depending on electric potential and field in anisotropic media.

Abstract

This paper investigates Calder\'on's problem on a conformally transversally anisotropic manifold $ (M,g) $ of dimension $n \geq 3$, where the conductivity $ a(s,x,p) $ might depend on both the electric potential and the electric field. We establish that for all $(t,x)\in \mathbb{R}\times M$ and $\beta \in \mathbb{N}^{1+n}$ the derivatives $ \partial_{(s,p)}^\beta a(s,x,p)|_{(s,p)=(t,0)}$ are uniquely determined by the boundary voltage-current measurements. If $ a(s,x,p) $ is analytic in $ p $, then $ a(s,x,p) $ can be uniquely recovered.