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

Registros recientes delgasterópodo exótico Rapana venosa (Valenciennes, 1846) a lo largode la línea costera Argentina: ¿Está progresandolainvasiónhaciaelsur?

Se presentan los registros más recientes de R. venosa a lo largo de la línea costera de Argentina (2010-2013), incluyendo el hallazgo de un individuo en la albufera Mar Chiquita (200 km al sur del Río de la Plata). Se discuten los posibles mecanismos de expansión.Recent range extensions of R. venosa along the Argentine coastline are presented (2010-2013),including the finding of an individual in Mar Chiquita lagoon (200 km southwards of the Río de la Plata).We discuss potential mechanisms for this expansion.Fil: Giberto, Diego Agustin. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Mar del Plata. Instituto de Investigaciones Marinas y Costeras. Subsede Instituto Nacional de Investigación y Desarrollo Pesquero; ArgentinaFil: Bruno, Luis Ignacio. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata; Argentina. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales; 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