Nonlinear time-harmonic Maxwell equations in domains
TLDR
In this paper, the authors give an introduction to the problem and the variational approach, and to survey recent results on ground and bound state solutions, including refinements of known results and some new results.Abstract:
The search for time-harmonic solutions of nonlinear Maxwell equations in the absence of charges and currents leads to the elliptic equation $$\begin{aligned} \nabla \times \left( \mu (x)^{-1}\nabla \times u\right) - \omega ^2\varepsilon (x)u = f(x,u) \end{aligned}$$
for the field $$u:\Omega \rightarrow \mathbb {R}^3$$
in a domain $$\Omega \subset \mathbb {R}^3$$
. Here, $$\varepsilon (x) \in \mathbb {R}^{3\times 3}$$
is the (linear) permittivity tensor of the material, and $$\mu (x) \in \mathbb {R}^{3\times 3}$$
denotes the magnetic permeability tensor. The nonlinearity $$f:\Omega \times \mathbb {R}^3\rightarrow \mathbb {R}^3$$
comes from the nonlinear polarization. If $$f=\nabla _uF$$
is a gradient, then this equation has a variational structure. The goal of this paper is to give an introduction to the problem and the variational approach, and to survey recent results on ground and bound state solutions. It also contains refinements of known results and some new results.read more
Citations
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Nonlinear time-harmonic Maxwell equations in an anisotropic bounded medium
Thomas Bartsch,Jarosław Mederski +1 more
TL;DR: In this paper, the authors studied the problem of finding a cylindrical symmetric solution to the time-harmonic electric field in an anisotropic material with a magnetic permeability tensor and a permittivity tensor.
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Uncountably Many Solutions for Nonlinear Helmholtz and Curl-Curl Equations
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The Brezis–Nirenberg problem for the curl–curl operator
TL;DR: In this paper, the authors studied the problem of finding a cylindrically symmetric ground state solution for the time-harmonic electric field in a nonlinear isotropic material with λ = − μ e ω 2 ≤ 0.
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Multiple solutions to a nonlinear curl-curl problem in $\mathbb{R}^3$
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Uncountably many solutions for nonlinear Helmholtz and curl-curl equations with general nonlinearities
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