1 Lectures at ICMat / UAM

We will consider the inverse conductivity problem or Calderón problem [11] in three dimensions (and higher). The fundamental paper is the work of Sylvester and Uhlmann [23] which gives global uniqueness in the inverse conductivity problem in three dimensions. In this problem we want to find a coefficient in an elliptic differential operator The first step will be to recover the coefficient at the boundary. Boundary identifiability has been considered by a number of authors including Kohn and Vogelius [15], Sylvester and Uhlmann [24], Kang and Yun [14]. [1] My work gives identifiability for conductivities that have no smoothness [7]. With Salo, we applied this technique to identify the coefficient of a first order term [10]. After the discussion of boundary identifiability, we will discuss the method of Sylvester and Uhlmann paying close attention to the regularity hypotheses needed to carry out the argument. The results we discuss will include work of Brown [8], Panchenko, Päivärinta and Uhlmann [18] and Brown and Torres [9]. We mention, but will not discuss, a recent interesting paper of Horst [13] gives stability for conductivities with 3/2 derivatives. These results require the conductivity to have 3/2 derivatives and no one believes that this is optimal. In two dimensions, Astala and Päivärinta [2] have given a method for recovering the conductivity that is only bounded and measurable. A construction due to Tolmasky [25], see also the work of Panchenko, Päivärinta and Uhlmann [18] allows us to construct exponentially growing solutions. We will give a version of this construction for the conductivity equation and indicate why this is not enough to allow recovery of the conductivity.