Extremal functions for the anisotropic Sobolev inequalities

The existence of multiple nonnegative solutions to the anisotropic critical problem - \sum_{i=1}^{N} \frac{\partial}{\partial x_i} (| \frac{\partial u}{\partial x_i} |^{p_i-2} \frac{\partial u}{\partial x_i}) = |u|^{p^*-2} u {in} \mathbb{R}^N is proved in suitable anisotropic Sobolev spaces. The solutions correspond to extremal functions of a certain best Sobolev constant. The main tool in our study is an adaptation of the well-known concentration-compactness lemma of P.-L. Lions to anisotropic operators. Futhermore, we show that the set of nontrival solutions \calS is included in $L^\infty(\R^N)$ and is located outside of a ball of radius $\tau >0$ in $L^{p^*}(\R^N)$

Associated factors of precocious sexual intercourse among schooled teenagers in Antsirabe town, Madagascar

Background: Teenagers are defined by world health organization as persons between 10 and 19 years of age. When this generation has a sexual intercourse, it is considered to be early. The main aim of this investigation is to identify associated factors of precocious sexual intercourse.Methods: A cross-sectional study was conducted at the high schools in Antsirabe town among teenagers.Results: Among 636 teenagers, 19.8% are prematurely initiated. The median age of first sexual experience is 16 years of age. The average age for this first sexual experience takes place at 15.5 (1.4) years of age for boys and at 16.6 (1.2) years for girls. One kind of sociodemographic profile is associated to the first precocious sexual intercourse. Poor school performance, urban life, alcohol, tobacco and drug use are also significantly associated with this precocious sexual experience. From the relationship standpoint, 6= lack of sexual education by the head of household, the absence of religious diligence, the influence of customs, internet access and accession a social network are indeed associated to this problem.Conclusions: In order to meet these results, the ministry of public health should design on Facebook, education program about forward sexuality

Protected area surface extension in Madagascar: Do endemism and threatened species remain useful criteria for site selection ?

The ‘hotspot approach’ considers that endemism and threatened species are key factors in protected area designation. Three wetland and forest sites have been proposed to be included into Madagascar’s system of protected areas (SAPM – Système des Aires Protégées de Madagascar). These sites are Manambolomaty (14,701 ha) and Mandrozo (15,145 ha) in the west and Bemanevika (37,041 ha) in the north. Biodiversity inventories of these three sites recorded 243 endemic species comprised of 44 reptiles, 54 amphibians, 104 birds, 23 small mammals, 17 lemurs and one fish. Of these 243 species, 30 are threatened taxa comprising two Critically Endangered (CR), 11 Endangered (EN) and 17 Vulnerable (VU) species. The long term ecological viability of these sites has been shown by population stability of the two Critically Endangered flagship species, the Madagascar fish eagle (Haliaeetus vociferoides) in Manambolomaty and Mandrozo and the recently rediscovered Madagascar pochard (Aythya innotata) in Bemanevika. Other threatened species and high biological diversity also justifies their inclusion into Madagascar’s SAPM

Hardy's inequality for functions vanishing on a part of the boundary

We develop a geometric framework for Hardy's inequality on a bounded domain when the functions do vanish only on a closed portion of the boundary.Comment: 26 pages, 2 figures, includes several improvements in Sections 6-8 allowing to relax the assumptions in the main results. Final version published at http://link.springer.com/article/10.1007%2Fs11118-015-9463-

Porous medium equation with nonlocal pressure

We provide a rather complete description of the results obtained so far on the nonlinear diffusion equation $u_t=\nabla\cdot (u^{m-1}\nabla (-\Delta)^{-s}u)$, which describes a flow through a porous medium driven by a nonlocal pressure. We consider constant parameters $m>1$ and $0<s<1$, we assume that the solutions are non-negative, and the problem is posed in the whole space. We present a theory of existence of solutions, results on uniqueness, and relation to other models. As new results of this paper, we prove the existence of self-similar solutions in the range when $N=1$ and $m>2$, and the asymptotic behavior of solutions when $N=1$. The cases $m = 1$ and $m = 2$ were rather well known.Comment: 24 pages, 2 figure

On the Finite Energy Weak Solutions to a System in Quantum Fluid Dynamics

In this paper we consider the global existence of weak solutions to a class of Quantum Hydrodynamics (QHD) systems with initial data, arbitrarily large in the energy norm. These type of models, initially proposed by Madelung, have been extensively used in Physics to investigate Supefluidity and Superconductivity phenomena and more recently in the modeling of semiconductor devices . Our approach is based on various tools, namely the wave functions polar decomposition, the construction of approximate solution via a fractional steps method, which iterates a Schr\"odinger Madelung picture with a suitable wave function updating mechanism. Therefore several \emph{a priori} bounds of energy, dispersive and local smoothing type allow us to prove the compactness of the approximating sequences. No uniqueness result is provided

Comparison of Marine Spatial Planning Methods in Madagascar Demonstrates Value of Alternative Approaches

The Government of Madagascar plans to increase marine protected area coverage by over one million hectares. To assist this process, we compare four methods for marine spatial planning of Madagascar's west coast. Input data for each method was drawn from the same variables: fishing pressure, exposure to climate change, and biodiversity (habitats, species distributions, biological richness, and biodiversity value). The first method compares visual color classifications of primary variables, the second uses binary combinations of these variables to produce a categorical classification of management actions, the third is a target-based optimization using Marxan, and the fourth is conservation ranking with Zonation. We present results from each method, and compare the latter three approaches for spatial coverage, biodiversity representation, fishing cost and persistence probability. All results included large areas in the north, central, and southern parts of western Madagascar. Achieving 30% representation targets with Marxan required twice the fish catch loss than the categorical method. The categorical classification and Zonation do not consider targets for conservation features. However, when we reduced Marxan targets to 16.3%, matching the representation level of the “strict protection” class of the categorical result, the methods show similar catch losses. The management category portfolio has complete coverage, and presents several management recommendations including strict protection. Zonation produces rapid conservation rankings across large, diverse datasets. Marxan is useful for identifying strict protected areas that meet representation targets, and minimize exposure probabilities for conservation features at low economic cost. We show that methods based on Zonation and a simple combination of variables can produce results comparable to Marxan for species representation and catch losses, demonstrating the value of comparing alternative approaches during initial stages of the planning process. Choosing an appropriate approach ultimately depends on scientific and political factors including representation targets, likelihood of adoption, and persistence goals

On a nonlocal stationary free-boundary problem arising in the confinement of a plasma in a stellarator geometry

We prove the existence and some qualitative properties of the solution to a two-dimensional free-boundary problem modeling the magnetic confinement of a plasma in a Stellarator configuration. The nonlinear elliptic partial differential equation on the plasma region was obtained from the three-dimensional magnetohydrodynamic system by HENDER & CARRERAS in 1984 by using averaging arguments and a suitable system of coordinates (Boozer's vacuum coordinates). The free boundary represents the separation between the plasma and vacuum regions, and the model is described by an inverse-type problem (some nonlinear terms of the equation are unknown). Using the zero net current condition for the Stellarator configurations, we reformulate the problem with the help of the notion of relative rearrangement, leading to a new problem involving nonlocal terms in the equation. We use an iterative algorithm and establish some new properties on the relative rearrangement in order to prove the convergence of the algorithm and then the existence of a solution