Spectral ℝ-Linear Problems: Applications to Complex Permittivity of Coated Cylinders
Zhunussova Z. Mityushev V.
June 2025Multidisciplinary Digital Publishing Institute (MDPI)
Mathematics
2025#13Issue 11
A composite-coated inclusion is embedded in a matrix, where the conductivity (permittivity) of the phases is assumed to be complex-valued. The purpose of this paper is to demonstrate that a non-zero flux can arise under specific conditions related to the conductivities of the components in the absence of external sources. These conditions are unattainable with conventional positive conductivities but can be satisfied when the conductivities are negative or complex—a scenario achievable in the context of metamaterials. The problem is formulated as a spectral boundary value problem for the Laplace equation, featuring a linear conjugation condition defined on a smooth curve L. This curve divides the plane (Formula presented.) into two regions, (Formula presented.) and (Formula presented.). The spectral parameter appears in the boundary condition, drawing parallels with the Steklov eigenvalue problem. The case of a circular annulus is analyzed using the method of functional equations. The complete set of eigenvalues is derived by applying the classical theory of self-adjoint operators in Hilbert space.
coated inclusion , complex-valued permittivity , functional equation , spectral boundary value problem
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Institute of Mathematics and Mathematical Modeling, Pushkina St., 125, Almaty, 050010, Kazakhstan
Faculty of Mechanics and Mathematics, Al-Farabi Kazakh National University, Al-Farabi Av., 71, Almaty, 050040, Kazakhstan
Faculty of Computer Technology and Cybersecurity, International Information Technology University, Manasa St., 34/1, Almaty, 050040, Kazakhstan
Faculty of Computer Science and Telecommunications, Cracow University of Technology, Warszawska St., 24, Krakow, 31-155, Poland
Institute of Mathematics and Mathematical Modeling
Faculty of Mechanics and Mathematics
Faculty of Computer Technology and Cybersecurity
Faculty of Computer Science and Telecommunications
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