Quasilinear equations with natural growth in the gradients in spaces of Sobolev multipliers

We study the existence problem for a class of nonlinear elliptic equations whose prototype is of the form \(-\Delta _p u = |\nabla u|^p + \sigma \) in a bounded domain \(\Omega \subset \mathbb {R}^n\). Here \(\Delta _p\), \(p>1\), is the standard p-Laplacian operator defined by \(\Delta _p u=\mathrm{div}\, (|\nabla u|^{p-2}\nabla u)\), and the datum \(\sigma \) is a signed distribution in \(\Omega \). The class of solutions that we are interested in consists of functions \(u\in W^{1,p}_0(\Omega )\) such that \(|\nabla u|\in M(W^{1,p}(\Omega )\rightarrow L^p(\Omega ))\), a space pointwise Sobolev multipliers consisting of functions \(f\in L^{p}(\Omega )\) such that

$$\begin{aligned} \int _{\Omega } |f|^{p} |\varphi |^p dx \le C \int _{\Omega } (|\nabla \varphi |^p + |\varphi |^p) dx \quad \forall \varphi \in C^\infty (\Omega ), \end{aligned}$$

for some \(C>0\). This is a natural class of solutions at least when the distribution \(\sigma \) is nonnegative and compactly supported in \(\Omega \). We show essentially that, with only a gap in the smallness constants, the above equation has a solution in this class if and only if one can write \(\sigma =\mathrm{div}\, F\) for a vector field F such that \(|F|^{\frac{1}{p-1}}\in M(W^{1,p}(\Omega )\rightarrow L^p(\Omega ))\). As an important application, via the exponential transformation \(u\mapsto v=e^{\frac{u}{p-1}}\), we obtain an existence result for the quasilinear equation of Schrödinger type \(-\Delta _p v = \sigma \, v^{p-1}\), \(v\ge 0\) in \(\Omega \), and \(v=1\) on \(\partial \Omega \), which is interesting in its own right.

     

ortho-Selective ethylation of phenol with ethanol catalyzed by bimetallic mesoporous catalyst, CoAl-MCM-41

CoAl-MCM-41 (X) catalysts with X = n_Si/(n_Co + n_Al) various ratios were synthesized and ethylation of phenol with ethanol was studied in vapor-phase at temperatures between 250 and 450 °C. The products obtained were O-alkylated product (ethyl phenyl ether), C-alkylated products (2-ethylphenol and 4-ethylphenol), and C-/O-alkylated products (ethyl ethylphenyl ether). The phenol conversion increased significantly with reaction temperature over all the catalysts. The activity of the catalysts followed the order CoAl-MCM-41 (20) > CoAl-MCM-41 (50) > CoAl-MCM-41 (80). Selectivity between the C-alkylation and the O-alkylation depended on the factors such as acidity of the catalyst and the reaction temperature. CoAl-MCM-41 (20) catalyst displayed a phenol conversion of 40% and a selectivity of more than 80% for 2-ethylphenol under the optimized reaction condition. The ethanol to phenol ratios and the reactant flow rate are also influential for both activity and selectivity of CoAl-MCM-41 catalysts.     

Axial development and radial non-uniformity of flow in packed columns

Flow inhomogeneity and axial development in low-pressure chromatographic columns have been studied by magnetic resonance imaging velocimetry. The columns studied included (a) an 11.7-mm I.D. column packed with either 50 microm diameter porous polyacrylamide, or 99 or 780 microm diameter impermeable polystyrene beads, and (b) a 5-mm I.D. column commercially packed with 10 microm polymeric beads. The packing methods included gravity settling, slurry packing, ultrasonication, and dry packing with vibration. The magnetic resonance method used averaged apparent fluid velocity over both column cross-sections and fluid displacements greater than one particle diameter and hence permits assessment of macroscopic flow non-uniformities. The results confirm that now non-uniformities induced by the conical distributor of the 11.7-mm I.D. column or the presence of voids at the column entrance relax on a length scale of the column radius. All of the 11.7-mm I.D. columns examined exhibit near wall channeling within a few particle diameters of the wall. The origins of this behavior are demonstrated by imaging of the radial dependence of the local porosity for a column packed with 780 microm beads. Columns packed with the 99-microm beads exhibit reduced flow in a region extending from ten to three-to-five particle diameters from the wall. This velocity reduction is consistent with a reduced porosity of 0.35 in this region as compared to approximately 0.43 in the bulk of the column. Ultrasonicated and dry-packed columns exhibit enhanced flow in a region located between approximately eight and 20 particle diameters from the wall. This enhancement maybe caused by packing density inhomogeneity and/or particle size segregation caused by vibration during the packing process. No significant non-uniformities on length scales of 20 microm or greater were observed in the commercially packed column packed with 10 microm particles.     

Absorbed dose calculations for macromolecular crystals: improvements to RADDOSE

Radiation damage is an unwelcome and unavoidable aspect of macromolecular crystallography. In order to quantify the extent of X‐ray‐induced changes, knowledge of the dose (absorbed energy per unit mass) is necessary since it is the obvious metric against which to plot variables such as diffraction intensity loss and B factors. Significant improvements to the program RADDOSE for accurately calculating the dose absorbed by macromolecular crystals are presented here. Specifically, the probability of energy loss through the escape of fluorescent photons from de‐excitation of an atom following photoelectric absorption is now included. For lighter elements, both the probability of fluorescence and of its subsequent escape from the crystal are negligible, but for heavier atoms the chance of fluorescence becomes significant (e.g. 30% as opposed to Auger electron decay from a K‐shell excited iron atom), and this has the effect of reducing the absorbed dose. The effects of this phenomenon on dose calculations are presented for examples of crystals of an iron‐containing protein, 2‐selenomethionine proteins, a uranium derivatised protein, and for a nucleic acid sample. For instance, the inclusion of fluorescent escape results in up to a 27% decrease in the calculated absorbed dose for a typical selenomethionine protein crystal irradiated at the selenium K‐edge.     

Isobutylene‐rich macromonomers: Dynamics and yields of peroxide‐initiated crosslinking

New isobutylene‐rich elastomers bearing multiple pendant styrenic, acrylic, maleimidic, vinylic, and allylic functional groups have been prepared and examined in the context of peroxide‐initiated crosslinking. Halide displacement from brominated poly(isobutylene‐co‐isoprene) (BIIR) by the requisite carboxylate nucleophiles in homogeneous toluene solutions provide the desired esters in quantitative yield without complications from dehydrohalogenation or premature crosslinking. Heating the resulting macromonomers with dicumyl peroxide to 160 °C under solvent‐free conditions gives thermoset derivatives, with reaction rates and yields depending markedly on functional group structure. In general, high cure extents can only be achieved using highly reactive pendant functional groups, owing to the competitive balance between crosslinking through C?C oligomerization, and degradation through β‐scission of backbone macroradical intermediates. Independent control of crosslinking rates and cure extents is gained through the use of nitroxyl radical traps bearing acrylate functionality. © 2014 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2015 , 53, 123–132     

Amino-functionalized MIL-101(Fe) metal-organic framework as a viable fluorescent probe for nitroaromatic compounds

Microchimica Acta - A highly luminescent iron(III)-based amino-functionalized metal-organic framework (MOF) of type NH2-MIL-101(Fe) was synthesized by a solvothermal method. Its structure and...