Related papers: Nonlinear time-harmonic Maxwell equations in domai…
Solutions of the linearized Vlasov-Poisson equations for the electric field radiated by a time varying point charge in a three-dimensional, unbounded, spatially homogeneous plasma with a uniform background magnetic field and a uniform…
In this paper we establish compactness results of multiscale and very weak multiscale type for sequences bounded in $L^{2}(0,T;H_{0}^{1}(\Omega ))$, fulfilling a certain condition. We apply the results in the homogenization of the parabolic…
Let us consider a semilinear boundary value problem $ - \Delta u= f(x,u),$ in $\Omega,$ with Dirichlet boundary conditions, where $ \Omega \subset \mathbb{R}^N $, $N> 2,$ is a bounded smooth domain. We provide sufficient conditions…
This paper investigates the existence and regularity of strong solutions to the incompressible Navier-Stokes equations within a bounded domain $\Omega \subset \mathbb{R}^3$, subject to the boundary condition $(u\cdot \vec{n})|_{\partial…
In our study we consider nonlinear, power-law field-dependent electrical permitivity and magnetic permeability and investigate the time-dependent Maxwell equations with the self-similar Ansatz. This is a first-order hyperbolic PDE system…
This work is devoted to the study of the boundary value problem \begin{eqnarray}\nonumber (-1)^\alpha \Delta^\alpha u = (-1)^k S_k[u] + \lambda f, \qquad x &\in& \Omega \subset \mathbb{R}^N, \\ \nonumber u = \partial_n u = \partial_n^2 u =…
We prove existence and multiplicity results for finite energy solutions to the nonlinear elliptic equation \[ -\triangle u+V\left( \left| x\right| \right) u=g\left( \left| x\right| ,u\right) \quad \textrm{in }\Omega \subseteq…
Complex formalism of Riemann - Silberstein - Majorana - Oppenheimer in Maxwell electrodynamics is extended to the case of arbitrary pseudo-Riemannian space - time in accordance with the tetrad recipe of Tetrode - Weyl - Fock - Ivanenko. In…
We prove in this paper the existence of solutions of nonlinear parabolic problems in inhomogeneous Musielak Orlicz Sobolev spaces, we assume neither a $\Delta_2$ nor $\nabla_2$ on the Musielak function $\varphi$. The main contribution of…
Consider the nonlinear parabolic equation in the form $$ u_t-{\rm div} \mathbf{a}(D u,x,t)={\rm div}\,(|F|^{p-2}F) \quad \text{in} \quad \Omega\times(0,T), $$ where $T>0$ and $\Omega$ is a Reifenberg domain. We suppose that the nonlinearity…
In this article, we study domains $\Omega \subset \mathbb{S}^2$ that support positive solutions of the overdetermined problem $$ \Delta u + f(u,|\nabla u|)=0 \quad \text{in } \Omega, $$ subject to the boundary conditions $u=0$ on…
In this work we establish existence and multiplicity of solutions for elliptic problem with nonlinear boundary conditions under strong resonance conditions at infinity. The nonlinearity is resonance at infinity and the reso- nance phenomena…
In this paper, we are concerned with the multiplicity of nontrivial solutions for the following class of complex problems $$ (-i\nabla - A(\mu x))^{2}u= \mu |u|^{q-2}u + |u|^{2^{*}-2}u \ \mbox{in} \ \Omega, \ \ \ \ u \in…
The Maxwell equations with accounting for tensors properties of time have been considered. The effects that follow from such consideration are described. These are the appearance of vacuum polarization, anisotropy of electromagnetic wave…
Due to the nonlinearity of QED, a static charge becomes a magnetic dipole if placed in a magnetic field. Already without external field, the cubic Maxwell equation for the field of a point charge has a soliton solution with a finite field…
We study the existence of solutions to the fractional elliptic equation (E1) $(-\Delta)^\alpha u+\epsilon g(|\nabla u|)=\nu $ in a bounded regular domain $\Omega$ of $\R^N (N\ge2)$, subject to the condition (E2) $u=0$ in $\Omega^c$, where…
Consider a Hamiltonian system of type \[ -\Delta u=H_{v}(u,v),\ -\Delta v=H_{u}(u,v) \ \ \text{ in } \Omega, \qquad u,v=0 \text{ on } \partial \Omega \] where $H$ is a power-type nonlinearity, for instance $H(u,v)= |u|^p/p+|v|^q/q$, having…
We are interested in looking for two nontrivial solutions for a class of nonhomogeneous quasilinear elliptic system with sign-changing weight functions and concave-convex nonlinearities on the bounded domain. This kind of quasilinear…
We develop a solution theory for a generalized electro-magneto static Maxwell system in an exterior domain with anisotropic coefficients converging at infinity with a certain rate towards the identity. Our main goal is to treat right hand…
The electric and magnetic fields of a spatio-temporally varying electric current loop are calculated using the Jefimenko equations. The radiation and the nonradiation parts of the electromagnetic fields are derived in the framework of…