Existence of solutions of a non-linear eigenvalue problem with a variable weight

We study the non-linear minimization problem on $H_0^1(\mathrm{\Omega })?L^q$ with $q=\frac{2n}{n?2}$, $\alpha >0$ and $n\ge 4$:

$\underset{\begin{array}{c}\hfill u\in H_0^1(\mathrm{\Omega })\hfill \\ \hfill {6u6}_{L^q}=1\hfill \end{array}}{\inf }?\underset{\mathrm{\Omega }}{\int }a(x,u)|\mathrm{?}u{|}^2?\lambda \underset{\mathrm{\Omega }}{\int }|u{|}^2$

where $a(x,s)$ presents a global minimum α at $(x_0,0)$ with $x_0\in \mathrm{\Omega }$. In order to describe the concentration of $u(x)$ around $x_0$, one needs to calibrate the behavior of $a(x,s)$ with respect to s. The model case is

$\underset{\begin{array}{c}\hfill u\in H_0^1(\mathrm{\Omega })\hfill \\ \hfill {6u6}_{L^q}=1\hfill \end{array}}{\inf }?\underset{\mathrm{\Omega }}{\int }(\alpha +|x{|}^{\beta }|u{|}^k)|\mathrm{?}u{|}^2?\lambda \underset{\mathrm{\Omega }}{\int }|u{|}^2.$

In a previous paper dedicated to the same problem with $\lambda =0$, we showed that minimizers exist only in the range $\beta <kn/q$, which corresponds to a dominant non-linear term. On the contrary, the linear influence for $\beta \ge kn/q$ prevented their existence. The goal of this present paper is to show that for $0<\lambda \le \alpha {\lambda }_1(\mathrm{\Omega })$, $0\le k\le q?2$ and $\beta >kn/q+2$, minimizers do exist.     

Theoretical analysis of the influence of the adsorption process on the voltammetric curves in the case of a second order adsorption step followed by a first order charge transfer reaction: Part III. Variation of the transfer coefficient and application to semiconductor electrodes in l.p.s.v. and s.s.c.v.

The kinetic behaviour of a two-step sequential reaction including a 2nd order adsorption process is treated theoretically in the case of semiconductor electrodes. It is shown that the electronic structure of the semiconductor can be taken into account in some simplified cases by introducing a formal transfer coefficient comprised between 0 and 1 and potential independent. In both linear potential sweep voltammetry and steady state cyclic voltammetry, the usual characteristic features (E_M, _M, i_M) of the voltammetric curves are calculated as a function of the sweep rate for different values of the transfer coefficient.     

An implicit high order finite volume scheme for the solution of 3D Navier–Stokes equations with new discretization of diffusive terms

The computation of three-dimensional viscous flows on complex geometries requiring distorted meshes is of great interest. This paper presents a finite volume solver using a quadratic reconstruction of the unknowns for the advective fluxes computation, and a conservative and consistent discretization of the diffusive terms, based on an extended version of the Coirier's diamond path. A fully implicit time integration procedure is employed with a preconditioned matrix-free GMRES solver.     

Preparation and reactivity of polyfunctional phenazine derivatives

Treatment of 2,3,6,7-tetramethylphenazine with N-bromosuccinimide under light irradiation afforded in good yield the tetrakis(bromomethyl) derivative. Weakly basic nucleophilic reagents such as thiols, phenols, amines and the azide anion allow the preparation of a variety of functionalized derivatives. Some 2,7-disubstituted phenazines have also been obtained.     

Theory of chirped photonic crystals

In chirped photonic crystals, the structural parameters describing a unit cell are progressively varied from a unit cell to the nearest ones. Geometric and dielectric response functions can be affected by this modulation, but here we only investigate the effect of a long-range, slowly varying, modulation of the refractive index. The Bloch modes are modified by essentially being modulated by an envelope function which adapts to the long-range dielectric function perturbation. It is shown that this envelope function obeys a simple linear Schrödinger equation of classical (non-quantum) origin. Close to a band extremum, at a gap edge, the envelope functions can be interpreted as wave functions of particles, called “energy carriers”. These carriers have a mass and come as two species, referred to as “effective photons” (for positive band curvatures) or “photonic holes” (for negative band curvatures). The energy transfer through the chirped structure can be viewed as resulting from the migration of these particles under forces implied by the long-range dielectric function modulation.     

L'eldanolide,phéromone des glandes alaires de la pyrale de la canne à sucre, eldanasaccharina (wlk.): structure et synthèse de ses deux énantiomères

The isolation and structure determination of eldanolide, the wing gland pheromone of the male African Sugar Cane Borer, Eldana saccharina (Wlk.) is described. The absolute configuration was determined as (3S, 4R) by comparison of the CD spectra of the natural pheromone with both synthetic enantiomers.     

Caro-Cryptands: Tris-carotenoid macrobicyclic ligands–synthesis,crystal structure,and dinuclear copper(I) complexes

The tris-carotenoid macrobicycles 1 and 2 were obtained in good yields in a one-step macrobicyclisation condensation between the tripode N(CH₂CH₂NH₂)₃ and the polyolefinic dialdehydes 5 and 6 . They form dinuclear cryptates by complexation of two Cu^I ions. The crystal structure of the tris-carotenoid compound 2 confirms that it contains three parallel polyolefinic strands. These substances may formally be considered as prototypes of molecular ‘cables’ formed by three electron-conducting molecular wires.