The inverse conductivity problem, first discussed by Calderón, consists of determining the conductivity of the interior of an object from measurements of electrical potential and current taken on the boundary. One aspect of this problem is uniqueness: whether or not two different conductivity functions might give rise to the same set of boundary measurements.

This question has been considered for different spatial dimensions and under various assumptions on the regularity of the conductivity function. In the case of two dimensions, [Brown and Uhlmann, 1997] established the uniqueness for conductivities in  with . [Astala and Päivärinta, 2006] proved the uniqueness for conductivities in . We prove a global uniqueness result for Calderón's problem in two dimensions with conductivities in . This condition allows for the conductivity to be unbounded.

One example would be $\gamma(x)=\log\left|\log|ax|\right|$. The choice of the Sobolev space  for the conductivity is optimal for the two dimensional Calderón's problem.