On some non-linear systems of PDEs related to inverse problems in 3-d conductivity

We focus on certain non-linear, non-convex, non-coercive systems of PDEs in three dimensions that are directly motivated by inverse problems in conductivity for the three-dimensional case. It turns out that such systems are variational, as they formally are the Euler-Lagrange systems associated with an explicit first-order functional, and thus we exploit both its variational structure as well as its connection to inverse problems. In particular, boundary conditions play a central role.