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