We study fine properties of the gradient of the solution to the conductivity equation div (σ-(x)∇u(x)) = 0 in bounded domains. Our analysis is restricted to dimension two and it is concerned with merely measurable, elliptic coefficients. We establish sharp results on the higher order of integrability of the modulus of the gradient and of the inverse of the modulus of the gradient of the solution with the aid of recent advances in the theory of quasiconformal mappings due to Astala, Eremenko and Hamilton. We also consider the first order system associated to the second order elliptic equation, hence defining the map w = (u, ũ) and we isolate a class of Dirichlet boundary data on the function u which guarantees the quasiconformality of the mapping w. This leads in particular to a geometrical characterization of the electrostatic energy. We make use of results about the critical points of solutions of elliptic equations due to Alessandrini and Alessandrini and Magnanini.
Leonetti, F., & Nesi, V. (1997). Quasiconformal solutions to certain first order systems and the proof of a conjecture of G. W. Milton. Journal Des Mathematiques Pures et Appliquees, 76(2), 109–124. https://doi.org/10.1016/S0021-7824(97)89947-3