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.

The inverse problem dealt with in this article consists of reconstructing the electrical conductivity from the free response of the system in the magneto-quasistationary (MQS) limit. The MQS limit corresponds to a diffusion PDE. In this framework, a key role is played by the monotonicity principle (MP), that is a monotone relation connecting the unknown material property to the (measured) free-response. The MP is relevant as the basis of noniterative and real-time imaging methods. The Monotonicity Principle has been found in many different physical problems governed by diverse PDEs. Despite its rather general nature, each physical/mathematical context requires the proper operator showing the MP to be identified. In order to achieve this, it is necessary to develop ad hoc mathematical approaches tailored to the specific framework. In this article, we prove that (i) there exists a monotonic relationship between the electrical resistivity and the time constants characterizing the free response for MQS systems and (ii) the induced current density can be represented through a modal expansion. These results are based on the analysis of an elliptic eigenvalue problem obtained from the separation of variables.

In this paper, we deal with an inverse electrical conductivity problem which considers the reconstruction of nonlinear electrical conductivity in steady currents operations using boundary measurements. In the current set up, we establish a monotonic relation between the unknown material property to the (measured) Dirichlet-to-Neumann operator (DtN). It is in fact the Monotonicity Principle which is the base of a class of non-iterative and real-time imaging methods and algorithms. To be more precise, we indicate the issues appear in our nonlinear case to transfer this Monotonicity result from the Dirichlet Energy to the DtN operator which is the fundamental huddle in comparison to linear and p-Laplacian cases. Finally, we introduce a new Average DtN operator which is different from the existing ones and resolves complications produced by non-linearity in our problem.

In this paper we study the first Steklov-Laplacian eigenvalue with an internal fixed spherichal obstacle. We prove that the spherical shell locally maximizes the first eigenvalue among nearly spherical sets when both the internal ball and the volume are fixed.

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