[Paper Review] Ground states of time-harmonic semilinear Maxwell equations in R^3 with vanishing permittivity

Jarosław Mederski|arXiv (Cornell University)|Jun 17, 2014

Advanced Mathematical Physics ProblemsMathematics21 references21 citations

TL;DR

This paper establishes the existence of ground state solutions to time-harmonic semilinear Maxwell equations in $ ^3$ with vanishing permittivity, using variational methods on a strongly indefinite functional. The key result shows that a ground state exists when the $L^{3/2}$-norm of the potential $V(x) = - ho au^2\varepsilon(x)$ is below the best Sobolev constant, under suitable growth conditions on the nonlinearity $F$.

ABSTRACT

We investigate the existence of solutions $E:\mathbb{R}^3 o\mathbb{R}^3$ of the time-harmonic semilinear Maxwell equation $$ abla imes( abla imes E) + V(x) E = \partial_E F(x,E) \quad ext{in}\mathbb{R}^3,$$ where $V:\mathbb{R}^3 o\mathbb{R}$, $V(x)\leq 0$ a.e. on $\mathbb{R}^3$, $ abla imes$ denotes the curl operator in $\mathbb{R}^3$ and $F:\mathbb{R}^3 imes\mathbb{R}^3 o\mathbb{R}$ is a nonlinear function in $E$. In particular we find a ground state solution provided that suitable growth conditions on $F$ are imposed and $L^{3/2}$-norm of $V$ is less than the best Sobolev constant. In applications $F$ is responsible for the nonlinear polarization and $V(x)=-μω^2\varepsilon(x)$ where $μ>0$ is the magnetic permeability, $ω$ is the frequency of the time-harmonic electric field $\Re\{E(x)e^{iωt}\}$ and $\varepsilon$ is the linear part of the permittivity in an inhomogeneous medium.

Motivation & Objective

• To establish the existence of ground state solutions for time-harmonic semilinear Maxwell equations in $ ^3$ with vanishing permittivity.
• To analyze the behavior of electromagnetic waves in epsilon-near-zero (ENZ) media where nonlinear polarization dominates.
• To address the challenge of a strongly indefinite variational structure due to the sign-changing potential $V(x) \leq 0$.
• To prove that the $L^{3/2}$-norm of $V$ being less than the best Sobolev constant ensures existence of a ground state.
• To investigate the physical relevance of such solutions in nonlinear optical media, particularly Kerr-like materials.

Proposed method

• Formulates the time-harmonic Maxwell equation as $\nabla \times (\nabla \times E) + V(x)E = \partial_E F(x,E)$ in $\r^3$, with $V(x) \leq 0$.
• Applies variational methods to a strongly indefinite functional associated with the equation, using the Nehari-Pankov manifold to locate critical points.
• Imposes growth conditions on $F$ such as $f(x,E) = \Gamma(x)\min\{|E|^{p-2}, |E|^{q-2}\}E$ with $2 < p \leq q$.
• Employs a concentration-compactness argument and global compactness analysis to overcome lack of compactness in $\r^3$.
• Uses a Pohozaev-type identity and scaling arguments to derive a Pohozaev identity for the solution, proving that the energy is finite and the solution is nontrivial.
• Applies a suitable truncation and approximation scheme to handle the nonlinearity and ensure the functional is well-defined on the space $\mathcal{D}(\mathrm{curl}, p, q)$.

Experimental results

Research questions

• RQ1 Under what conditions does a ground state solution exist for the time-harmonic semilinear Maxwell equation in $\r^3$ with vanishing permittivity?
• RQ2 How does the $L^{3/2}$-norm of the potential $V(x)$ relate to the existence of nontrivial solutions?
• RQ3 Can variational methods be applied to a strongly indefinite functional arising from Maxwell's equations with sign-changing potential?
• RQ4 What role does the nonlinearity $F$ play in ensuring the existence of localized, finite-energy solutions?
• RQ5 Is it possible to rule out nontrivial solutions under certain growth conditions on $F$ when $V = 0$ or $V$ is constant?

Key findings

• A ground state solution exists for the time-harmonic semilinear Maxwell equation in $\r^3$ if the $L^{3/2}$-norm of $V(x)$ is strictly less than the best Sobolev constant.
• The existence is established under the assumption that the nonlinearity $F$ satisfies suitable subcritical growth conditions, including the case $p=4$ corresponding to the Kerr effect.
• For $V=0$ or $V$ constant negative, no nontrivial classical solution exists if $q \leq 6$ or $p > 6$, due to a contradiction derived from the Pohozaev identity.
• The solution $E$ belongs to the space $\mathcal{D}(\mathrm{curl}, p, q)$, which embeds into $L^{p,q} = L^p(\r^3, \r^3) + L^q(\r^3, \r^3)$, ensuring finite electromagnetic energy.
• The energy functional satisfies the Pohozaev identity: $\int_{\r^3} |\nabla \times E|^2 dx = 3 \int_{\r^3} F(E) dx + \int_{\r^3} \langle f(E), E \rangle dx$, which is crucial for proving the existence of a ground state.
• The method successfully handles the lack of compactness in $\r^3$ via a global compactness argument and the Nehari-Pankov manifold technique, ensuring convergence of Palais-Smale sequences.