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.

我们研究了一些PDE的非线性系统，这些系统证明与电导率中的经典逆问题有关。从某种意义上说，它们具有变分结构，至少在形式上，它们是一些显式矢量变分问题的欧拉-拉格朗日系统。但是，底层的非负整数是非拟凸的，乍一看，相应的Euler-Lagrange系统的弱解的存在就受到了损害。然而，值得注意的是，可以非常显式地计算拟凸凸化。尽管我们的分析范围更广，但是当存在最小值消失的全局最小化器时，就建立了与反电导率问题的联系。由于此事实取决于边界条件，因此我们尝试通过拟凸化来澄清情况，并描述了一种构建合成边界数据的方法，对于该数据，基本功能已达到最低要求并消失了。因为这正是电导率反问题的情况，所以我们在数值上探索了逼近此类全局极小值的可能性，并通过典型的下降或牛顿-拉夫森方法为反问题提供了近似解。我们使用这些综合数据在几个示例上测试了该过程。