Some non-linear systems of PDEs related to inverse problems in conductivity

Abstract

We study some non-linear systems of PDEs that turn out to be related to the classical inverse problem in conductivity. They have a variational structure in the sense that, at least formally, they are the Euler–Lagrange systems of some explicit vector variational problems. The underlying, non-negative integrands are, however, non-quasiconvex in such a way that the existence of weak solutions for the corresponding Euler–Lagrange systems is, at first sight, compromised. It is however remarkable that the quasiconvexification can be computed quite explicitly. Though our analysis is broader, the connection with inverse conductivity problems is established when there are global minimizers with a vanishing minimum value. Since this fact depends on boundary conditions, we try to clarify the situation with the help of the quasiconvexification, and describe a way to build synthetic boundary data for which the minimum of the underlying functional is attained and vanishes. Because this is exactly the situation for inverse problems in conductivity, we explore numerically the possibility of approximating such global minimizers, furnishing approximated solutions to inverse problems through typical descent or Newton–Raphson methods. We test the procedure on several examples with such synthetic data.