Symmetrization for fractional Neumann problems

In this paper we complement the program concerning the application of symmetrization methods to nonlocal PDEs by providing new estimates, in the sense of mass concentration comparison, for solutions to linear fractional elliptic and parabolic PDEs with Neumann boundary conditions. These results are achieved by employing suitable symmetrization arguments to the Stinga-Torrea local extension problems, corresponding to the fractional boundary value problems considered. Sharp estimates are obtained first for elliptic equations and a certain number of consequences in terms of regularity estimates is then established. Finally, a parabolic symmetrization result is covered as an application of the elliptic concentration estimates in the implicit time discretization scheme.Comment: 34 page

Improved Poincar\'e inequalities

Although the Hardy inequality corresponding to one quadratic singularity, with optimal constant, does not admit any extremal function, it is well known that such a potential can be improved, in the sense that a positive term can be added to the quadratic singularity without violating the inequality, and even a whole asymptotic expansion can be build, with optimal constants for each term. This phenomenon has not been much studied for other inequalities. Our purpose is to prove that it also holds for the gaussian Poincar\'e inequality. The method is based on a recursion formula, which allows to identify the optimal constants in the asymptotic expansion, order by order. We also apply the same strategy to a family of Hardy-Poincar\'e inequalities which interpolate between Hardy and gaussian Poincar\'e inequalities

Bourgain-Brezis-Mironescu formula for magnetic operators

We prove a Brezis-Bourgain-Mironescu type formula for a class of nonlocal magnetic spaces, which builds a bridge between a fractional magnetic operator recently introduced and the classical theory.Comment: revised versio

Symmetrization for Linear and Nonlinear Fractional Parabolic Equations of Porous Medium Type

We establish symmetrization results for the solutions of the linear fractional diffusion equation $\partial_t u +(-\Delta)^{\sigma/2}u=f$ and itselliptic counterpart $h v +(-\Delta)^{\sigma/2}v=f$, $h>0$, using the concept of comparison of concentrations. The results extend to the nonlinear version, $\partial_t u+(-\Delta)^{\sigma/2}A(u)=f$, but only when A:\re_+\to\re_+ is a concave function. In the elliptic case, complete symmetrization results are proved for $\,B(v)+(-\Delta)^{\sigma/2}v=f$ \ when $B(v)$ is a convex nonnegative function for $v>0$ with $B(0)=0$, and partial results when $B$ is concave. Remarkable counterexamples are constructed for the parabolic equation when $A$ is convex, resp. for the elliptic equation when $B$ is concave. Such counterexamples do not exist in the standard diffusion case $\sigma=2$.Comment: 42 pages, 1 figur

Fractional semilinear Neumann problems arising from a fractional Keller--Segel model

We consider the following fractional semilinear Neumann problem on a smooth bounded domain $\Omega\subset\mathbb{R}^n$, $n\geq2$, $$\begin{cases} (-\varepsilon\Delta)^{1/2}u+u=u^{p},&\hbox{in}~\Omega,\\ \partial_\nu u=0,&\hbox{on}~\partial\Omega,\\ u>0,&\hbox{in}~\Omega, \end{cases}$$ where $\varepsilon>0$ and $1<p<(n+1)/(n-1)$. This is the fractional version of the semilinear Neumann problem studied by Lin--Ni--Takagi in the late 80's. The problem arises by considering steady states of the Keller--Segel model with nonlocal chemical concentration diffusion. Using the semigroup language for the extension method and variational techniques, we prove existence of nonconstant smooth solutions for small $\varepsilon$, which are obtained by minimizing a suitable energy functional. In the case of large $\varepsilon$ we obtain nonexistence of nonconstant solutions. It is also shown that as $\varepsilon\to0$ the solutions $u_\varepsilon$ tend to zero in measure on $\Omega$, while they form spikes in $\overline{\Omega}$. The regularity estimates of the fractional Neumann Laplacian that we develop here are essential for the analysis. The latter results are of independent interest

Structural Changes by Thermal Treatment up to Glass Obtention of P2O5-Na2O-CaO-SiO2 Compounds with Bioglass Composition Types

P2O5-Na2O-CaO-SiO2 compounds are the base of certain glass types. Glasses are solids obtained by fast cooling of melted mix of certain compounds. Different compositions give origin to many products with a variety of applications such as: bottles, coatings, windows, tools for chemical industry, laboratory equipment, optics, as bioceramics, etc. The aim of this work was to analyze structural changes of different composition in the P2O5-Na2O-CaO-SiO2 systems thermally treated up to 1250˚C, that is to say, before glass formation, by X ray diffraction. Intermediate and final developed phases up to 1100˚C thermal treatment in samples were generated as a function of Na2O/CaO (1 and 1.62) and P2O5/Na2O ratios (0, 0.2 and 0.245). High- and low-combeites, calcium and sodium-calcium silicate were found at the highest studied temperature.Fil: Volzone, Cristina. Provincia de Buenos Aires. Gobernación. Comision de Invest.científicas. Centro de Tecnología de Recursos Minerales y Ceramica. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - la Plata. Centro de Tecnología de Recursos Minerales y Ceramica; ArgentinaFil: Stábile, Franco Matías. Provincia de Buenos Aires. Gobernación. Comision de Invest.científicas. Centro de Tecnología de Recursos Minerales y Ceramica. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - la Plata. Centro de Tecnología de Recursos Minerales y Ceramica; Argentin

Symmetrization for fractional elliptic and parabolic equations and an isoperimetric application

We develop further the theory of symmetrization of fractional Laplacian operators contained in recent works of two of the authors. The theory leads to optimal estimates in the form of concentration comparison inequalities for both elliptic and parabolic equations. In this paper we extend the theory for the so-called \emph{restricted} fractional Laplacian defined on a bounded domain $\Omega$ of $\mathbb R^N$ with zero Dirichlet conditions outside of $\Omega$. As an application, we derive an original proof of the corresponding fractional Faber-Krahn inequality. We also provide a more classical variational proof of the inequality.Comment: arXiv admin note: substantial text overlap with arXiv:1303.297

Nanoclay as Adsorbent: Evaluation for Removing Dyes Used in the Textile Industry

The dyes commonly used in the textile industry have structural resonant valence distribution, within a complex molecule, which determines the color of the dyes. The coloring is produced by chromospheres holding radiation in UV-visible range. Besides coloring, these compounds can be found in waste water and cause serious problems in living organisms, because their biodegradation products may be a source of toxic substances such as amines, which are generated from the characteristic azo chromospheres groups in most of the dyes. Current treatments for color removal from hazardous waste from these materials are complex and costly, for this reason some industries do not treat the wastes generated during the staining step. The aim of this study is to evaluate the use of nanoclay as adsorbents for dye used in the textile industry. The dyes used in this work are commercially known as: Orange GR, Black GN an African Brown. Nanoclay was obtained by bentonite treated with organic cation. The retention of the dye was evaluated by measurements in the UV-visible spectrum by using a HP-8354 equipment.Fil: Martínez Stagnaro, Susana Yamila. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional del Comahue. Facultad de Ingeniería. Asentamiento Universidad Zapala; ArgentinaFil: Volzone, Cristina. Provincia de Buenos Aires. Gobernación. Comisión de Investigaciones Científicas. Centro de Tecnología de Recursos Minerales y Cerámica. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Centro de Tecnología de Recursos Minerales y Cerámica; ArgentinaFil: Huck, Lucas Ruben. Provincia de Buenos Aires. Gobernación. Comisión de Investigaciones Científicas. Centro de Tecnología de Recursos Minerales y Cerámica. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Centro de Tecnología de Recursos Minerales y Cerámica; Argentin

Sorption Water By Modified Bentonite

In order to improve adsorptive properties and provide a viable alternative to humidity control techniques, the desiccants can be modified with chemical aggregates such as calcium chloride (CaCl2), the latter often used as dehydrating due to its hygroscopicity and low cost. Its use as a drying agent has disadvantages because of its particles form a liquid film. However, the CaCl2 may be mixed with other materials for preparing desiccant products. In this work we studied, at 298 K and different relative humidities (RH), the adsorption processes of desiccants mixtures composed of a bentonite clay Argentinian and CaCl2. The results showed a better performance of the desiccant mixture with respect to natural clay in the whole HR working range. It was also observed that addition of CaCl2 favors adsorptive properties with respect to a purification treatment.Fil: Castrillo, Natalia Paola. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Salta. Instituto de Investigaciones para la Industria Química. Universidad Nacional de Salta. Facultad de Ingeniería. Instituto de Investigaciones para la Industria Química; Argentina. Universidad Catolica de Salta. Consejo de Investigaciones; ArgentinaFil: Mercado, Adela Isabel Guadalupe. Universidad Catolica de Salta. Consejo de Investigaciones; Argentina. Universidad Nacional de Salta. Facultad de Ciencias Exactas; ArgentinaFil: Volzone, Cristina. Provincia de Buenos Aires. Gobernación. Comisión de Investigaciones Científicas. Centro de Tecnología de Recursos Minerales y Cerámica. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Centro de Tecnología de Recursos Minerales y Cerámica; Argentin