Calder\'{o}n problem for the quasilinear conductivity equation in dimension $2$

In this paper we prove a uniqueness result for the Calder\'{o}n problem for the quasilinear conductivity equation on a bounded domain $\R^2$. The proof of the result is based on the higher order linearization method, which reduces the problem to showing density of products of solutions to the linearized equation and their gradients. In contrast to the higher dimensional case, the proof involves delicate analysis of the correction terms of Bukhgeim type complex geometric solutions (CGOs), which have only limited decay. To prove our results, we construct suitable families of CGOs whose phase functions have and do not have critical points. We also combine stationary phase analysis with $L^p$ estimates for the correction terms of the CGOs.