We treat an inverse electrical conductivity problem which deals with the reconstruction of nonlinear electrical conductivity starting from boundary measurements in steady currents operations. In this framework, a key role is played by the Monotonicity Principle, which establishes a monotonic relation connecting the unknown material property to the (measured) Dirichlet-to-Neumann operator (DtN). Monotonicity Principles are the foundation for a class of non-iterative and real-time imaging methods and algorithms. In this article, we prove that the monotonicity principle for the Dirichlet Energy in nonlinear problems holds under mild assumptions. Then, we show that apart from linear and p-Laplacian cases, it is impossible to transfer this monotonicity result from the Dirichlet Energy to the DtN operator. To overcome this issue, we introduce a new boundary operator, identified as an average DtN operator.

我们处理反电导率问题，该问题处理从稳态电流操作中的边界测量开始的非线性电导率的重建。在这个框架中，单调性原理发挥了关键作用，它建立了一个单调关系，将未知材料属性与（测量的）狄利克雷到诺依曼算子（DtN）联系起来。单调性原理是一类非迭代实时成像方法和算法的基础。在本文中，我们证明了非线性问题中狄利克雷能量的单调性原理在温和的假设下成立。然后，我们证明除了线性和p- 拉普拉斯算子，不可能将这个单调性结果从狄利克雷能量转移到 DtN 算子。为了克服这个问题，我们引入了一个新的边界算子，被识别为平均 DtN 算子。