A nonlinear hyperbolic Maxwell system using measure-valued functions

We consider a modified antenna's problem with power-type constitutive laws. This consists in a new nonlinear hyperbolic system that extends a Duvaut-Lions model. Using the Galerkin approximation, properties of the natural functional spaces, and exploring the $L^p$-$L^{p'}$ duality, we prove the existence of solutions, in a generalized sense, passing to the limit in a family of approximated problems and using measure-valued functions. In this process the difficulties in obtaining the necessary a priori estimates for the solutions of the finite-dimensional problems are overcome through the use of bases with special properties related to the model.The research of the authors was partially supported by the Research Centre of Mathematics of the University of Minho through the FCT Pluriannual Funding Program

Multiplicador de Lagrange num problema com restrição não constante no gradiente

Prova-se a existência de solução, num sentido generalizado, de um problema com um multiplicador de Lagrange, para uma restrição arbitrária no gradiente e condição de Dirichlet homogénea na fronteira. Prova-se ainda a equivalência deste problema com a correspondente inequação variacional elíptica. A abordagem utilizada para provar o resultado de existência baseia-se na utilização de soluções de uma família aproximante de equações quasilineares elípticas

A class of electromagnetic p-curl systems: blow-up and finite time extinction

We study a class of $p$-curl systems arising in electromagnetism, for $\frac65 < p < \infty$, with nonlinear source or sink terms. Denoting by $\boldsymbol h$ the magnetic field, the source terms considered are of the form $\boldsymbol h\left(\int_\Omega|\boldsymbol h|^2\right)^{\frac{\sigma-2}{2}}$, with $\sigma\geq1$. Existence of local or global solutions is proved depending on values of $\sigma$ and $p$. The blow-up of local solutions is also studied. The sink term is of the form $\boldsymbol h\left(\int_\Omega|\boldsymbol h|^k\right)^{-\lambda}$, with $k,\lambda>0$. Existence and finite time extinction of solutions are proved, for certain values of $k$ and $\lambda$.The first author was supported partially by the research project PTDC/MAT/110613/2009, FCT, Portugal. The research of the second and third authors was partially supported by CMAT-"Centro de Matemetica da Universidade do Minho", financed by FEDER Funds through "Programa Operacional Factores de Competitividade-COMPETE'' and by Portuguese Funds through FCT- "Funda ao para a Ciencia e a Tecnologia", within the Project Est-C/MAT/UI0013/2011

Blow-up and finite time extinction for p(x, t)-curl systems arising in electromagnetism

"Available online 22 March 2016"We study a class of $p(x,t)$-curl systems arising in electromagnetism, with a nonlinear source term. Denoting by $\boldsymbol{h}$ the magnetic field, the source term considered is of the form $\lambda\boldsymbol{h}\left( \int_{\Omega}|\boldsymbol{h}|^{2}\right)^{\frac{\sigma-2}{2}}$ where $\lambda\in\{-1,0,1\}$: when $\lambda\in\{-1,0\}$ we consider $0<\sigma\leq2$ and for $\lambda=1$ we have $\sigma\geq1$. We introduce a suitable functional framework and a convenient basis that allow us to apply the Galerkin's method and prove existence of local or global solutions, depending on the values of $\lambda$ and $\sigma$. We study the finite time extinction or the stabilization towards zero of the solutions when $\lambda\in\{-1,0\}$ and the blow-up of local solutions when $\lambda=1$.The research was partially supported by the Research Center CMAF-CIO of the University of Lisbon, Portugal, by the Research Center CMAT of the University of Minho, Portugal, with the Portuguese Funds from the "Fundacao para a Ciencia e a Tecnologia", through the Projects UID/MAT/04561/2013 and PEstOE/MAT/UI0013/2014, respectively, and by the Grant No. 15-11-20019 of the Russian Science Foundation, Russia

A class of stationary nonlinear Maxwell systems

We study a new class of electromagnetostatic problems in the variational framework of the subspace of $W^{1,p}(\Omega)^3$ of vector functions with zero divergence and zero normal trace, for $p>6/5$, in smooth, bounded and simply connected domains $\Omega$ of $\mathbb R^3$. We prove a Poincaré-Friedrichs type inequality and we obtain the existence of steady-state solutions for an electromagnetic induction heating problem and for a quasi-variational inequality modelling a critical state generalized problem for type-II superconductors

On a p-curl system arising in electromagnetism

We prove existence of solution of a $p$-curl type evolutionary system arising in electromagnetism with a power nonlinearity of order $p$, $1<p<\infty$, assuming natural tangential boundary conditions. We consider also the asymptotic behaviour in the power obtaining, when $p$ tends to infinity, a variational inequality with a curl constraint. We also discuss the existence, uniqueness and continuous dependence on the data of the solutions to general variational inequalities with curl constraints dependent on time, as well as the asymptotic stabilization in time towards the stationary solution with and without constraint.The first an last authors are supported by the Research Centre of Mathematics of the University of Minho through the FCT Pluriannual Funding Program and FCT project UT-Austin/MAT/0035/2008

Wannier-Bloch approach to localization in high harmonics generation in solids

Emission of high-order harmonics from solids provides a new avenue in attosecond science. On one hand, it allows to investigate fundamental processes of the non-linear response of electrons driven by a strong laser pulse in a periodic crystal lattice. On the other hand, it opens new paths toward efficient attosecond pulse generation, novel imaging of electronic wave functions, and enhancement of high-order harmonic generation (HHG) intensity. A key feature of HHG in a solid (as compared to the well-understood phenomena of HHG in an atomic gas) is the delocalization of the process, whereby an electron ionized from one site in the periodic lattice may recombine with any other. Here, we develop an analytic model, based on the localized Wannier wave functions in the valence band and delocalized Bloch functions in the conduction band. This Wannier-Bloch approach assesses the contributions of individual lattice sites to the HHG process, and hence addresses precisely the question of localization of harmonic emission in solids. We apply this model to investigate HHG in a ZnO crystal for two different orientations, corresponding to wider and narrower valence and conduction bands, respectively. Interestingly, for narrower bands, the HHG process shows significant localization, similar to harmonic generation in atoms. For all cases, the delocalized contributions to HHG emission are highest near the band-gap energy. Our results pave the way to controlling localized contributions to HHG in a solid crystal, with hard to overestimate implications for the emerging area of atto-nanoscience

Intracellular localization of Crimean-Congo Hemorrhagic Fever (CCHF) virus glycoproteins

BACKGROUND: Crimean-Congo Hemorrhagic Fever virus (CCHFV), a member of the genus Nairovirus, family Bunyaviridae, is a tick-borne pathogen causing severe disease in humans. To better understand the CCHFV life cycle and explore potential intervention strategies, we studied the biosynthesis and intracellular targeting of the glycoproteins, which are encoded by the M genome segment. RESULTS: Following determination of the complete genome sequence of the CCHFV reference strain IbAr10200, we generated expression plasmids for the individual expression of the glycoproteins G(N )and G(C), using CMV- and chicken β-actin-driven promoters. The cellular localization of recombinantly expressed CCHFV glycoproteins was compared to authentic glycoproteins expressed during virus infection using indirect immunofluorescence assays, subcellular fractionation/western blot assays and confocal microscopy. To further elucidate potential intracellular targeting/retention signals of the two glycoproteins, GFP-fusion proteins containing different parts of the CCHFV glycoprotein were analyzed for their intracellular targeting. The N-terminal glycoprotein G(N )localized to the Golgi complex, a process mediated by retention/targeting signal(s) in the cytoplasmic domain and ectodomain of this protein. In contrast, the C-terminal glycoprotein G(C )remained in the endoplasmic reticulum but could be rescued into the Golgi complex by co-expression of G(N). CONCLUSION: The data are consistent with the intracellular targeting of most bunyavirus glycoproteins and support the general model for assembly and budding of bunyavirus particles in the Golgi compartment