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Time‐harmonic and asymptotically linear Maxwell equations in anisotropic media
Author(s) -
Qin Dongdong,
Tang Xianhua
Publication year - 2017
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4615
Subject(s) - mathematics , nehari manifold , mathematical analysis , maxwell's equations , homogeneous space , lipschitz domain , bounded function , anisotropy , tensor (intrinsic definition) , boundary value problem , lipschitz continuity , nonlinear system , pure mathematics , geometry , physics , quantum mechanics
This paper is focused on following time‐harmonic Maxwell equation:∇ × ( μ − 1 ( x ) ∇ × u ) − ω 2 ε ( x ) u = f ( x , u ) ,in Ω ,ν × u = 0 ,on ∂ Ω ,where Ω ⊂ R 3 is a bounded Lipschitz domain, ν : ∂ Ω → R 3 is the exterior normal, and ω is the frequency. The boundary condition holds when Ω is surrounded by a perfect conductor. Assuming that f is asymptotically linear as | u | → ∞ , we study the above equation by improving the generalized Nehari manifold method. For an anisotropic material with magnetic permeability tensor μ ∈ R 3 × 3and permittivity tensor ε ∈ R 3 × 3 , ground state solutions are established in this paper. Applying the principle of symmetric criticality, we find 2 types of solutions with cylindrical symmetries in particular for the uniaxial material.