A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient

We consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient, such as

$?F(x,u,Du,D^{2}u)=\lambda c(x)u+〈M(x)Du,Du〉+h(x)$

in a bounded domain with a Dirichlet boundary condition; here $\lambda \in \mathbb{R}$, $c,h\in L^p(\mathrm{\Omega })$, $p>n\ge 1$, $c?0$ and the matrix M satisfies $0<{\mu }_{1}I\le M\le {\mu }_{2}I$. Recently this problem was studied in the “coercive” case $\lambda c\le 0$, where uniqueness of solutions can be expected; and it was conjectured that the solution set is more complex for noncoercive equations. This conjecture was verified in 2015 by Arcoya, de Coster, Jeanjean and Tanaka for equations in divergence form, by exploiting the integral formulation of the problem. Here we show that similar phenomena occur for general, even fully nonlinear, equations in nondivergence form. We use different techniques based on the maximum principle.We develop a new method to obtain the crucial uniform a priori bounds, which permit to us to use degree theory. This method is based on basic regularity estimates such as half-Harnack inequalities, and on a Vázquez type strong maximum principle for our kind of equations.     

Simultaneous quantification of levofloxacin,pefloxacin, ciprofloxacin and moxifloxacin in microvolumes of human plasma using high‐performance liquid chromatography with ultraviolet detection

Levofloxacin, pefloxacin, ciprofloxacin and moxifloxacin are four fluoroquinolones used in the treatment of serious bacterial infections. The antibacterial activity of fluoroquinolones is concentration dependent. Therefore, therapeutic drug monitoring in daily clinical practice is warranted to ensure the therapy's efficacy and prevent bacterial resistance. The purpose of the present study was to develop a method using high‐pressure liquid chromatography with an ultraviolet detector for simultaneous quantification of these four fluoroquinolones in human plasma. A 50 μL aliquot of plasma was precipitated by 200 μL of methanol using gatifloxacin as internal standard. The chromatographic separation was performed on a Kinetex XB‐C₁₈ column using a mobile phase composed of a mixture of orthophosphoric acid 0.4% (v/v), acetonitrile and methanol at a flow rate of 1.2 mL/min. Dual UV wavelength mode was used, with levofloxacin and moxifloxacin monitored at 293 nm, and pefloxacin and ciprofloxacin monitored at 280 nm. The calibration was linear over the ranges of 0.125–25 mg/L for levofloxacin, 0.1–20mg/L for moxifloxacin and 0.05‐10 mg/L for both pefloxacin and ciprofloxacin. Inter‐ and intra‐day trueness and precision were <13% for all the compounds under study. The proposed method was simple, reliable, cost‐effective and suitable for therapeutic drug monitoring or pharmacokinetics studies.     

A Dual Threat: Redox-Activity and Electronic Structures of Well-Defined Donor–Acceptor Fulleretic Covalent-Organic Materials

The effect of donor (D)–acceptor (A) alignment on the materials electronic structure was probed for the first time using novel purely organic porous crystalline materials with covalently bound two- and three-dimensional acceptors. The first studies towards estimation of charge transfer rates as a function of acceptor stacking are in line with the experimentally observed drastic, eight-fold conductivity enhancement. The first evaluation of redox behavior of buckyball- or tetracyanoquinodimethane-integrated crystalline was conducted. In parallel with tailoring the D-A alignment responsible for “static” changes in materials properties, an external stimulus was applied for “dynamic” control of the electronic profiles. Overall, the presented D–A strategic design, with stimuli-controlled electronic behavior, redox activity, and modularity could be used as a blueprint for the development of electroactive and conductive multidimensional and multifunctional crystalline porous materials.     

Confinement-Driven Photophysics in Hydrazone-Based Hierarchical Materials

Confinement-imposed photophysics was probed for novel stimuli-responsive hydrazone-based compounds demonstrating a conceptual difference in their behavior within 2D versus 3D porous matrices for the first time. The challenges associated with photoswitch isomerization arising from host interactions with photochromic compounds in 2D scaffolds could be overcome in 3D materials. Solution-like photoisomerization rate constants were realized for sterically demanding hydrazone derivatives in the solid state through their coordinative immobilization in 3D scaffolds. According to steady-state and time-resolved photophysical measurements and theoretical modeling, this approach provides access to hydrazone-based materials with fast photoisomerization kinetics in the solid state. Fast isomerization of integrated hydrazone derivatives allows for probing and tailoring resonance energy transfer (ET) processes as a function of excitation wavelength, providing a novel pathway for ET modulation.     

Clustering Six Electrons within “Dawson-Like” Polyoxometalate: An Open Route toward Its Post-functionalization

Super-reduction of polyoxometalates (POMs) in solution is of fundamental interest for designing innovative energy storage systems. In this article, we show that the “Dawson-like” POM can undergo a disproportionation process during its massive electron uptake, leading to species containing three metal-metal bonds as evidenced by X-ray diffraction, multi-nuclear magnetic resonance spectroscopy (¹H and ¹⁸³W NMR), extended X-ray absorption fine structure (EXAFS), UV/Vis, and voltammetry techniques. This result indicates that electron storing within metal-metal bonds is not a unique property of Keggin-type POM as postulated since the 70s. Besides, we demonstrate that the presence of an electron-rich triad in the “Dawson-like” POM allows its post-functionalization with additional tungstate ions, generating a chiral molecule that is also the largest W^IV-containing POMs known to date.     

Facility Update: APS Operations and New Beamlines

Twenty years is a long time, but it seems to me as if it were yesterday when the three of us—Michel Belakhovsky, Christian Riekel and I—first scientists of the Experiments Division (except the directors) at the not yet created European Synchrotron Radiation Facility, living in a temporary and not very solid building, made the last corrections to the text of the famous “Red Book,” the ESRF Foundation Phase Report published in February 1987. The floor was shaking in our office when Ruprecht Haensel, the first Director General, walked along the corridor…. We had convened workshops of experts in the various fields to define the first set of beamlines to be built at the ESRF. The program was quite complete, comprising inelastic scattering, imaging (topography), microfocusing, small angle scattering, nuclear resonance, high energy, etc. Costs were particularly difficult to estimate, but finally we were not so far from the real figures.     

Carbon Dots (C‐dots) from Cow Manure with Impressive Subcellular Selectivity Tuned by Simple Chemical Modification

Improved cellular selectivity for nucleoli staining was achieved by simple chemical modification of carbon dots (C‐dots) synthesized from waste carbon sources such as cow manure (or from glucose). The C‐dots were characterized and functionalized (amine‐passivated) with ethylenediamine, affording amide bonds that resulted in bright green fluorescence. The new modified C‐dots were successfully applied as selective live‐cell fluorescence imaging probes with impressive subcellular selectivity and the ability to selectively stain nucleoli in breast cancer cell lineages (MCF‐7). The C‐dots were also tested in four other cellular models and showed the same cellular selection in live‐cell imaging experiments.     

Solvatomorphism in (E)‐2‐(2,6‐dichloro‐4‐hydroxybenzylidene)hydrazinecarboximidamide

The structures of orthorhombic (E)‐4‐(2‐{[amino(iminio)methyl]amino}vinyl)‐3,5‐dichlorophenolate dihydrate, C₈H₈Cl₂N₄O·2H₂O, (I), triclinic (E)‐4‐(2‐{[amino(iminio)methyl]amino}vinyl)‐3,5‐dichlorophenolate methanol disolvate, C₈H₈Cl₂N₄O·2CH₄O, (II), and orthorhombic (E)‐amino[(2,6‐dichloro‐4‐hydroxystyryl)amino]methaniminium acetate, C₈H₉Cl₂N₄O⁺·C₂H₃O₂⁻, (III), all crystallize with one formula unit in the asymmetric unit, with the molecule in an E configuration and the phenol H atom transferred to the guanidine N atom. Although the molecules of the title compounds form extended chains via hydrogen bonding in all three forms, owing to the presence of different solvent molecules, those chains are connected differently in the individual forms. In (II), the molecules are all coplanar, while in (I) and (III), adjacent molecules are tilted relative to one another to varying degrees. Also, because of the variation in hydrogen‐bond‐formation ability of the solvents, the hydrogen‐bonding arrangements vary in the three forms.     

Supramolecular Systems from Natural Polymers and Maleic Polyelectrolytes

Summary: The formation of polyelectrolyte complexes by interaction between chitosan and maleic acid copolymers as strong/weak dibasic polyanions was investigated. The interaction between the sodium salt of maleic acid copolymers with styrene or vinyl acetate and the chitosan hydrochloride in aqueous solution was followed by potentiometric, conductometric and turbidimetric titration. The effect of the added low molecular salt on the complex formation was also investigated. The precipitated complexes were analyzed by FTIR spectroscopy and TG analysis. Preliminary layer-by-layer deposition experiments were performed to obtain thin films.