Cloaking for a quasi-linear elliptic partial differential equation

2017/04/10 by Tuhin Ghosh, Karthik Iyer · 1 citation
Engineering · Mathematics · Physics and Astronomy · #Boundary value problem #Bounded function #Cloak #Cloaking #Electromagnetic Scattering and Analysis #Elliptic partial differential equation #Helmholtz equation #Homogenization (climate) #Isotropy #Mathematical analysis #Mathematics #Metamaterial #Numerical methods in engineering #Numerical methods in inverse problems #Partial differential equation #Physics #Quantum mechanics #Sobolev space #math.AP

paper · pdf · doi:10.3934/ipi.2018020

published in Inverse Problems and Imaging 12(2), 461-491 (American Institute of Mathematical Sciences)

arxiv created 2017/04/10 · openalex publication_date 2018/01/01 · arxiv updated 2019/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

In this article we consider cloaking for a quasi-linear elliptic partial differential equation of divergence type defined on a bounded domain in ℝ^N for N = 2, 3. We show that a perfect cloak can be obtained via a singular change of variables scheme and an approximate cloak can be achieved via a regular change of variables scheme. These approximate cloaks, though non-degenerate, are anisotropic. We also show, within the framework of homogenization, that it is possible to get isotropic regular approximate cloaks. This work generalizes to quasi-linear settings previous work on cloaking in the context of Electrical Impedance Tomography for the conductivity equation.

Citations

• Ideal Cylindrical Cloak: Perfect but Sensitive to Tiny Perturbations
• Elliptic Partial Differential Equations of Second Order
• Determining anisotropic real‐analytic conductivities by boundary measurements
• Invisibility and Inverse Problems
• Global Uniqueness for a Two-Dimensional Inverse Boundary Value Problem
• A Global Uniqueness Theorem for an Inverse Boundary Value Problem
• Homogenization of Differential Operators and Integral Functionals