Related papers: Nonlinear time-harmonic Maxwell equations in an an…

We find solutions $E:\Omega\to\mathbb{R}^3$ of the problem \[ \left\{\begin{aligned} &\nabla\times(\nabla\times E) + \lambda E = \partial_E F(x,E) &&\quad \text{in}\Omega\\ &\nu\times E = 0 &&\quad \text{on}\partial\Omega \end{aligned}…

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…

We look for solutions $E:\Omega\to\mathbb{R}^3$ of the problem $$ \left\{ \begin{aligned} &\nabla\times(\nabla\times E) +\lambda E = |E|^{p-2}E &&\quad \text{in }\Omega &\nu\times E = 0 &&\quad \text{on }\partial\Omega \end{aligned} \right.…

We investigate the existence of solutions $E:\mathbb{R}^3\to\mathbb{R}^3$ of the time-harmonic semilinear Maxwell equation $$\nabla\times(\nabla\times E) + V(x) E = \partial_E F(x,E) \quad \text{in}\mathbb{R}^3,$$ where…

We survey recent results concerning ground states and bound states $u\colon\mathbb{R}^3\to\mathbb{R}^3$ to the curl-curl problem $$\nabla\times(\nabla\times u)+V(x)u= f(x,u) \quad\hbox{ in } \mathbb{R}^3,$$ which originates from the…

We consider a quasilinear nonhomogeneous, anisotropic Maxwell system in a bounded smooth domain of $\mathbb{R}^{3}$ with a strictly positive conductivity subject to the boundary conditions of a perfect conductor. Under appropriate…

We consider the time-harmonic Maxwell equations posed in $\mathbb{R}^3$. We prove a priori bounds on the solution for $L^\infty$ coefficients $\epsilon$ and $\mu$ satisfying certain monotonicity properties, with these bounds valid for…

We are interested in the nonlinear, time-harmonic Maxwell equation $$ \nabla \times (\nabla \times \mathbf{E} ) + V(x) \mathbf{E} = h(x, \mathbf{E})\mbox{ in } \mathbb{R}^3 $$ with sign-changing nonlinear term $h$, i.e. we assume that $h$…

We look for travelling wave fields $$ E(x,y,z,t)= U(x,y) \cos(kz+\omega t)+ \widetilde U(x,y)\sin(kz+\omega t),\quad (x,y,z)\in\mathbb{R}^3,\, t\in\mathbb{R} $$ satisfying Maxwell's equations in a nonlinear medium which is not necessarily…

In this paper, we develop a new representation for outgoing solutions to the time harmonic Maxwell equations in unbounded domains in $\bbR^3.$ This representation leads to a Fredholm integral equation of the second kind for solving the…

We solve time-harmonic Maxwell's equations in anisotropic, spatially homogeneous media in intersections of $L^p$-spaces. The material laws are time-independent. The analysis requires Fourier restriction-extension estimates for perturbations…

We look for ground states and bound states $E:\mathbb{R}^3\to\mathbb{R}^3$ to the curl-curl problem $$\nabla\times(\nabla\times E)= f(x,E) \qquad\hbox{in } \mathbb{R}^3$$ which originates from nonlinear Maxwell equations. The energy…

We prove that the time-harmonic solutions to Maxwell's equations in a 3D exterior domain converge to a certain static solution as the frequency tends to zero. We work in weighted Sobolev spaces and construct new compactly supported…

We analyze the behavior of third-order in time linear and nonlinear sound waves in thermally relaxing fluids and gases as the sound diffusivity vanishes. The nonlinear acoustic propagation is modeled by the Jordan--Moore--Gibson--Thompson…

We study a general nonlinear parabolic equation on a Lipschitz bounded domain in $\mathbb{R}^N$, \begin{equation*} \left\{\begin{array}{l l} \partial_t u-\mathrm{div} A(t,x,\nabla u)= f(t,x)&\text{in}\ \ \Omega_T,\\ u(t,x)=0 &\ \mathrm{ on}…

When anisotropy is involved, the wave equation becomes simultaneous partial differential equations that are not easily solved. Moreover, when the anisotropy occurs due to both permittivity and permeability, these equations are insolvable…

We review Maxwell's equations and constitutive relations for 3D bianisotropic media in a generalized form: we consider all four variables and allow for nonzero polarization or magnetization, and also nonzero nonzero magnetic charge or…

We study the time harmonic Maxwell equations in a meta-material consisting of perfect conductors and void space. The meta-material is assumed to be periodic with period $\eta > 0$; we study the behaviour of solutions $(E^{\eta}, H^{\eta})$…

In this paper, we study the existence of solutions for a critical time-harmonic Maxwell equation in nonlocal media. By introducing some suitable Coulomb spaces involving curl operator, we are able to obtain the ground state solutions of the…

A dynamical Maxwell system is \begin{align*} & e_t={\rm curl\,} h, \quad h_t=-{\rm curl\,} e &&{\rm in}\,\,\Omega \times (0,T) & e|_{t=0}=0,\,\,\,\,h|_{t=0}=0 &&{\rm in}\,\,\Omega & e_\theta =f &&{\rm in}\,\,\, \partial\Omega \times [0,T]…