Some non-linear systems of PDEs related to inverse problems in conductivity
The main focus of the article is the Euler-Lagrange system associated with the functional \[ I(u) = \int_{\Omega} |\nabla u_1| |\nabla u_2| \mathrm{d} x. \] This gives a system of PDEs with a connection to the Calderón's problem: the conductivity will be \[ \gamma = \frac{\|\nabla u_2\|}{\|\nabla u_1\|}. \] The functional is non-coercive and not quasiconvex, foiling the direct method in calculus of variations. The authors present a modified problem with is non-negative and zero precisely at solutions and consider its quasiconvexification. This problem has a connection to a boundary condition for \(u_1\) and \(u_2\). Numerical tests investigate the functioning of Newton-Raphson method for solving the nonlinear system of equations.
scientific article; zbMATH DE number 7355288
scientific article