Nonlinear time-harmonic Maxwell equations in an anisotropic bounded medium

TL;DR

This paper establishes the existence of solutions for nonlinear time-harmonic Maxwell equations in anisotropic media, including ground and bound states, with special solutions for uniaxial materials exhibiting cylindrical symmetry.

Contribution

It introduces new existence results for nonlinear Maxwell equations with anisotropic, superquadratic nonlinearities, including symmetric solutions in uniaxial media.

Findings

Existence of ground state solutions.

Existence of bound states when nonlinearity is even.

Special solutions with cylindrical symmetry in uniaxial materials.

Abstract

We find solutions $E:\Omega\to\mathbb{R}^3$ of the problem \begin{eqnarray*} \left\{ \begin{aligned} &\nabla\times(\mu(x)^{-1}\nabla\times E) - \omega^2\epsilon(x) E = \partial_E F(x,E) &&\quad \text{in }\Omega\\%\newline &\nu\times E = 0 &&\quad \text{on }\partial\Omega \end{aligned} \right. \end{eqnarray*} on a bounded Lipschitz domain $\Omega\subset\mathbb{R}^3$ with exterior normal $\nu:\partial\Omega\to\mathbb{R}^3$. Here $\nabla\times$ denotes the curl operator in $\mathbb{R}^3$. The equation describes the propagation of the time-harmonic electric field $\Re\{E(x)e^{i\omega t}\}$ in an anisotropic material with a magnetic permeability tensor $\mu(x)\in\mathbb{R}^{3\times3}$ and a permittivity tensor $\epsilon(x)\in\mathbb{R}^{3\times3}$. The boundary conditions are those for $\Omega$ surrounded by a perfect conductor. It is required that $\mu(x)$ and $\epsilon(x)$ are symmetric and positive definite uniformly for $x\in\Omega$, and that $\mu,\epsilon\in L^{\infty}(\Omega,\mathbb{R}^{3\times 3})$. The nonlinearity $F:\Omega\times\mathbb{R}^3\to\mathbb{R}$ is superquadratic and subcritical in $E$, the model nonlinearity being of Kerr-type: $F(x,E)=|\Gamma(x)E|^p$ for some $2<p<6$ with $\Gamma(x)\in GL(3)$ invertible for every $x\in\Omega$ and $\Gamma,\Gamma^{-1}\in L^\infty(\Omega, \mathbb{R}^{3\times 3})$. We prove the existence of a ground state solution and of bound states if $F$ is even in $E$. Moreover if the material is uniaxial we find two types of solutions with cylindrical symmetries.