Nonlinear time-harmonic Maxwell equations in domains

TL;DR

This paper introduces the nonlinear time-harmonic Maxwell equations in domains, explores their variational structure, and surveys recent results on solutions, including some new findings and refinements of existing results.

Contribution

It provides an introduction to the variational approach for nonlinear Maxwell equations and surveys recent advances, including new results and refinements.

Findings

Survey of recent results on ground and bound state solutions

Refinements of known variational results

Introduction of variational methods for nonlinear Maxwell equations

Abstract

The search for time-harmonic solutions of nonlinear Maxwell equations in the absence of charges and currents leads to the elliptic equation $$\nabla\times\left(\mu(x)^{-1} \nabla\times u\right) - \omega^2\varepsilon(x)u = f(x,u)$$ for the field $u:\Omega\to\mathbb{R}^3$ in a domain $\Omega\subset\mathbb{R}^3$. Here $\varepsilon(x) \in \mathbb{R}^{3\times3}$ is the (linear) permittivity tensor of the material, and $\mu(x) \in \mathbb{R}^{3\times3}$ denotes the magnetic permeability tensor. The nonlinearity $f:\Omega\times\mathbb{R}^3\to\mathbb{R}^3$ 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.