Nonlinear singular elliptic problem with gradient term and general datum

In this paper we deal with an elliptic problem, with a first order term in natural growth, a singular nonlinearity and a L¹$L^1$ datum. We obtain an existence result without any smallness condition on the datum. We prove also a multiplicity result of positive solutions under additional conditions on the data.     

Structure–property study of new [1,2,3]-triazole liquid crystalline derivatives

The synthesis of a new type of liquid crystalline compound derived from 1,2,3-triazole was achieved using Wittig reaction. Polarised microscopy studies, X-ray scattering and differential scanning calorimetry analysis revealed that the target compounds exhibit enantiotropic liquid crystalline properties. Their mesomorphic behaviour is closely related to the nature of substituents in position N1 of the heterocycle, the length of the mesogenic core as well as the isomerisation nature (E/Z) of the double bond.     

Titanium-mediated addition of diallylsilanes to oxalyl chloride: formation of a diquinane

The titanium tetrachloride mediated reaction of the cis,cis-2,6-dimethyl-1,8-bis(trimethylsilyl)-2,6-octadiene with oxalyl chloride yielded 1-hydroxy-4-methyl-8-(1-propen-2-yl)bicyclo[3.3.0]oct-3-en-2-one of which the structure was confirmed by X-ray crystallographic analysis.     

Some remarks on systems of elliptic equations doubly critical in the whole $${\mathbb{R}^N}$$

We study the existence of different types of positive solutions to problem

AES, EELS and TRIM investigation of InSb and InP compounds subjected to Ar ions bombardment

The interaction of ions with matter plays an important role in the treatment of material surfaces. In this paper we study the effect of argon ion bombardment on the InSb surface in comparison with the InP one. The Ar⁺ ions, accelerated at low energy (300 eV) lead to compositional and structural changes in InP and InSb compounds. The InP surface is more sensitive to Ar⁺ ions than that of InSb. These results are directly inferred from the qualitative Auger electron spectra (AES) and electron energy loss spectroscopy (EELS) analysis. However, these techniques alone do not allow us to determine with accuracy the disturbed depth in Ar⁺ ions of InP and InSb compounds. For this reason, we combine AES and EELS with the simulation method TRIM (transport and range of ions in matter) to show the mechanism of interaction between the ions and the InP or InSb and hence determine the disturbed depth as a function of Ar⁺ energy.     

A remark on the fractional Hardy inequality with a remainder term

We prove in this note the following sharpened fractional Hardy inequality:     

A semilinear parabolic problem with singular term at the boundary

In this paper, we deal with a class of semilinear parabolic problems related to a Hardy inequality with singular weight at the boundary.

More precisely, we consider the problem

$$\left\{\begin{array}{l@{\quad}l}u_t-\Delta u=\lambda \frac{u^p}{d^2}&\text{ in }\,\Omega_{T}\equiv\Omega \times (0,T), \\u>0 &\text{ in }\,{\Omega_T}, \\u(x,0)=u_0(x)>0 &\text{ in }\,\Omega, \\u=0 &\text{ on }\partial \Omega \times (0,T),\end{array}\right.$$

(P)

where Ω is a bounded regular domain of \({\mathbbm{R}^N}\), \({d(x)=\text{dist}(x,\partial\Omega)}\), \({p > 0}\), and \({\lambda > 0}\) is a positive constant.

We prove that

1. 1.
   If \({0 < p < 1}\), then (P) has no positive very weak solution.
    
2. 2.
   If \({p=1}\), then (P) has a positive very weak solution under additional hypotheses on \({\lambda}\) and \({u_0}\).
    
3. 3.
   If \({p > 1}\), then, for all \({\lambda > 0}\), the problem (P) has a positive very weak solution under suitable hypothesis on \({u_0}\).
    

Moreover, we consider also the concave–convex-related case.

     

Ti-catalyzed reactions of 4,4′-bis(trimethylsilyl)bicyclohexyl-2,2′-diene with various electrophiles

Reductive disilylation (Li + Me₃SiCl − THF) of 1,3-cyclohexadiene led to 4,4′-bis(trimethylsilyl)bicyclohexyl-2,2′-diene (1). In the presence of TiCl₄ in dichloromethane, 1 reacted with some acyl chlorides, anhydrides, and aldehydes to give tricyclo[7.4.0.0^3,8]trideca-4,12-diene-2-yl derivatives.     

Renormalized solutions to nonlinear parabolic problems with blowing up coefficients and general measure data

An existence result is established for a class of quasilinear parabolic problem which is a diffusion type equations having continuous coefficients blowing up for a finite value of the unknown, a second hand \(\mu \in \mathcal {M}_{b}(Q)\) and an initial data \(u_{0}\in L^{1}(\Omega )\). We develop a technique which relies on the notion of a renormalized solution and an adequate regularization in time for certain truncation functions. Some compactness results are also shown under additional hypotheses.

     

AES, LEED and PYS investigation of Au deposits on InSe/Si(1 1 1) substrate

Auger Electron Spectroscopy (AES), Low Energy Electron Diffraction (LEED) and Photoelectron Yield Spectroscopy (PYS) measurements have been used to monitor the interaction of gold (Au) deposits on InSe/Si(1 1 1) substrate. Au has been sequentially deposed under ultra-high vacuum onto 40 Å-thick film of layered semiconductor InSe which is epitaxially grown by molecular beam epitaxy (MBE) on a Si(1 1 1)1 × 1-H substrate and kept at room temperature. Au coverage varies from 0.5 monolayer to 20 monolayers (ML) (in terms of InSe atomic surface plane: 1 ML = 7.2 1014 at/cm²) which is corresponding to 1.30 Å of Au-metal. The Au/InSe/Si(1 1 1) system was characterized as function of Au deposit, we noticed an interaction at room temperature starts as an apparent intercalation process until 5 ML. Beyond this dose Au islands begin to form on the sample surface without interaction with InSe substrate, thus the interface is far from to be a simple junction Au-InSe.