On the second Dirichlet eigenvalue of some nonlinear anisotropic elliptic operators

Pietra, Francesco Della; Gavitone, Nunzia; Piscitelli, Gianpaolo

doi:10.1016/j.bulsci.2019.02.005

Cited by 17 publications

(5 citation statements)

References 20 publications

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“…There exist many papers dedicated to the study of its eigenvalues, for different boundary conditions (Dirichlet, Neumann, Robin or Steklov). See, e.g., [13], [20], [21], [22], [25], [32], [44] and references therein.…”

Section: )mentioning

confidence: 99%

On eigenvalue problems governed by the (p,q)-Laplacian

Barbu

Moroşanu²

2023

Stud. Univ. Babes-Bolyai Math.

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Section: )mentioning

confidence: 99%

On eigenvalue problems governed by the (p,q)-Laplacian

Barbu

Moroşanu²

2023

Stud. Univ. Babes-Bolyai Math.

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“…Eventually, we stress that problems, as in (2.8), have been broadly studied in various fields of mathematics (see e.g. [83][84][85] and reference therein).…”

Section: Well-posedness Of the Forward Problemmentioning

confidence: 99%

Monotonicity Principle in tomography of nonlinear conducting materials ^*

et al. 2021

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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.

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“…Here we characterize higher eigenvalues (and eigenfunctions) for problem (3.1) by the means of an iterative process. Various characterizations are possible for higher eigenvalues (see [60,61] and reference therein). By induction, we define τ n (η) := max…”

Section: The Higher Eigenvaluesmentioning

confidence: 99%

The monotonicity principle for magnetic induction tomography

Tamburrino¹,

Piscitelli²,

Zhou³

2021

Inverse Problems

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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.

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On the second Dirichlet eigenvalue of some nonlinear anisotropic elliptic operators

Cited by 17 publications

References 20 publications

On eigenvalue problems governed by the (p,q)-Laplacian

On eigenvalue problems governed by the (p,q)-Laplacian

Monotonicity Principle in tomography of nonlinear conducting materials ^*

The monotonicity principle for magnetic induction tomography

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On the second Dirichlet eigenvalue of some nonlinear anisotropic elliptic operators

Cited by 17 publications

References 20 publications

On eigenvalue problems governed by the (p,q)-Laplacian

On eigenvalue problems governed by the (p,q)-Laplacian

Monotonicity Principle in tomography of nonlinear conducting materials *

The monotonicity principle for magnetic induction tomography

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Monotonicity Principle in tomography of nonlinear conducting materials ^*