Estimates on the Green’s Function and Existence of Positive Solutions of Nonlinear Singular Elliptic Equations in the Half Space

We establish a new 3G-Theorem for the Green’s function for the half space We exploit this result to introduce a new class of potentials that we characterize by means of the Gauss semigroup on . Next, we define a subclass of and we study it. In particular, we prove that properly contains the classical Kato class . Finally, we study the existence of positive continuous solutions in of the following nonlinear elliptic problem

Sur la convergence en moyenne des suites presque sousadditives

Sans résumé     

Morphological variation in the oral disc of the scleractinian coral Favia speciosa (Dana) at Indonesia

Photographic analysis was used to examine morphological differences in the oral disc of n=1196 living polyps of Favia speciosa Dana (1846) sampled from four sites in the Wakatobi Marine National Park, Indonesia. Although oral disc size attributes differed significantly between the study sites, the geographic difference accounted for only a small fraction of the morphological variation and did not show a clear pattern of correspondence to sedimentation rates. A much higher fraction of the morphological variation was attributed to depth and so to incident light: oral discs grew significantly larger with increasing depth. These results suggest that for F. speciosa corals at Wakatobi, oral disc size may be optimised for heterotrophic nutrition under low light conditions, and photosynthesis in conditions where light is not limiting. Furthermore, the driving force for this phenotypic plasticity is more likely to be depth than sedimentation rate.     

Properties of Fixed Point Sets of Monotone Nonexpansive Mappings in Banach Spaces

In this work, we discuss the properties of the common fixed points set of a commuting family of monotone nonexpansive mappings. In particular, we show that under suitable assumptions, this set is a monotone nonexpansive retract.     

Characterization of the Existence of an Enriched Linear Finite Element Approximation Using Biorthogonal Systems

The present paper is intended to give a characterization of the existence of an enriched linear finite element approximation based on biorthogonal systems. It is shown that the enriched element exists if and only if a certain multivariate generalized trapezoidal type cubature formula has a nonzero approximation error. Furthermore, for such an enriched element we derive simple explicit formulas for the basis functions, and show that the approximation error can be written as the sum of the error of the (non-enriched) element plus a perturbation that depends on the enrichment function. Finally, we estimate the approximation error in L² norm. We also give an alternative approach to estimate the approximation error, which relies on an appropriate use of the Poincaré inequality.     

A geometric property in and its applications

In this work, we initiate the study of the geometry of the variable exponent sequence space when . In 1931 Orlicz introduced the variable exponent sequence spaces while studying lacunary Fourier series. Since then, much progress has been made in the understanding of these spaces and of their continuous counterpart. In particular, it is well known that is uniformly convex if and only if the exponent is bounded away from 1 and infinity. The geometry of when either or remains largely ill‐understood. We state and prove a modular version of the geometric property of when , known as uniform convexity in every direction. We present specific applications to fixed point theory. In particular we obtain an analogue to the classical Kirk's fixed point theorem in when .     

Regeneration of expanded graphite electrodes by joined electrochemical and ozone treatment in liquid phase

The researches presented in this work were devoted to electrochemico-chemical regeneration of exhausted electrode made of expanded graphite (EG). The aimed process was conducted by electrochemical treatment and ozone flow performed together in wet environment. EG was covered with insoluble products of incomplete oxidation of phenol formed during cyclic voltammetry measurement. The same electrochemical technique was applied for evaluation of regeneration efficiency. To understand the process of EG regeneration, the electrode was characterized by calculating of BET surface, FTIR and XPS analysis. Moreover, SEM images of the investigated samples were also done. Obtained results have showed the success of regeneration treatment, which led to significant enhancement of electrode activity compared to original EG. The present work also revealed that the mechanism of phenol electrooxidation is changed after the regeneration treatment of electrode material. This effect is probably caused by the modification of chemical composition of EG surface due to its interactions with OH radicals intensively generated during the process of regeneration.

     

Estimates on the Green function and existence of positive solutions for some polyharmonic nonlinear equations in the half space

Let G_m,n be the Green function of with Dirichlet boundary conditions We establish some estimates on G_m,n, including a 3G-Theorem. Next, we introduce a Kato class of functions and we exploit properties of these functions to study the existence of positive solutions of some m-polyharmonic nonlinear elliptic problems.Mathematics Subject Classification (2000): 34B27, 35J40     

Reactions of chemically produced O2(1Δ) with Pb atoms

The potential of the chemical O₂(¹Δ) generator as a means of electronic excitation of metal atoms and oxides is demonstrated in a study of the reactions of O₂(¹Δ) with Pb. The resonant energy transfer process, Pb(³P_o) + O₂(¹Δ) → Pb(³P₁)+ O₂(³ν-), is found to be the key for producing excited Pb and PbO.