Singular solutions of fully nonlinear elliptic equations and applications

We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in each cone of $\mathbb{R}^n$, and the solutions are unique in an appropriate sense. We introduce a new method for analyzing the behavior of solutions near certain Lipschitz boundary points, which permits us to classify isolated boundary singularities of solutions which are bounded from either above or below. We also obtain a sharp Phragm\'en-Lindel\"of result as well as a principle of positive singularities in certain Lipschitz domains.Comment: 41 pages, 2 figure

Existence and multiplicity for elliptic problems with quadratic growth in the gradient

We show that a class of divergence-form elliptic problems with quadratic growth in the gradient and non-coercive zero order terms are solvable, under essentially optimal hypotheses on the coefficients in the equation. In addition, we prove that the solutions are in general not unique. The case where the zero order term has the opposite sign was already intensively studied and the uniqueness is the rule.Comment: To appear in Comm. PD

Positive Least Energy Solutions and Phase Separation for Coupled Schrodinger Equations with Critical Exponent: Higher Dimensional Case

We study the following nonlinear Schr\"{o}dinger system which is related to Bose-Einstein condensate: {displaymath} {cases}-\Delta u +\la_1 u = \mu_1 u^{2^\ast-1}+\beta u^{\frac{2^\ast}{2}-1}v^{\frac{2^\ast}{2}}, \quad x\in \Omega, -\Delta v +\la_2 v =\mu_2 v^{2^\ast-1}+\beta v^{\frac{2^\ast}{2}-1} u^{\frac{2^\ast}{2}}, \quad x\in \om, u\ge 0, v\ge 0 \,\,\hbox{in \om},\quad u=v=0 \,\,\hbox{on \partial\om}.{cases}{displaymath} Here \om\subset \R^N is a smooth bounded domain, $2^\ast:=\frac{2N}{N-2}$ is the Sobolev critical exponent, -\la_1(\om)0 and $\beta\neq 0$, where \lambda_1(\om) is the first eigenvalue of $-\Delta$ with the Dirichlet boundary condition. When \bb=0, this is just the well-known Brezis-Nirenberg problem. The special case N=4 was studied by the authors in (Arch. Ration. Mech. Anal. 205: 515-551, 2012). In this paper we consider {\it the higher dimensional case $N\ge 5$}. It is interesting that we can prove the existence of a positive least energy solution (u_\bb, v_\bb) {\it for any $\beta\neq 0$} (which can not hold in the special case N=4). We also study the limit behavior of (u_\bb, v_\bb) as $\beta\to -\infty$ and phase separation is expected. In particular, u_\bb-v_\bb will converge to {\it sign-changing solutions} of the Brezis-Nirenberg problem, provided $N\ge 6$. In case \la_1=\la_2, the classification of the least energy solutions is also studied. It turns out that some quite different phenomena appear comparing to the special case N=4.Comment: 48 pages. This is a revised version of arXiv:1209.2522v1 [math.AP

Symbiotic Bright Solitary Wave Solutions of Coupled Nonlinear Schrodinger Equations

Conventionally, bright solitary wave solutions can be obtained in self-focusing nonlinear Schrodinger equations with attractive self-interaction. However, when self-interaction becomes repulsive, it seems impossible to have bright solitary wave solution. Here we show that there exists symbiotic bright solitary wave solution of coupled nonlinear Schrodinger equations with repulsive self-interaction but strongly attractive interspecies interaction. For such coupled nonlinear Schrodinger equations in two and three dimensional domains, we prove the existence of least energy solutions and study the location and configuration of symbiotic bright solitons. We use Nehari's manifold to construct least energy solutions and derive their asymptotic behaviors by some techniques of singular perturbation problems.Comment: to appear in Nonlinearit

Fibers and global geometry of functions

Since the seminal work of Ambrosetti and Prodi, the study of global folds was enriched by geometric concepts and extensions accomodating new examples. We present the advantages of considering fibers, a construction dating to Berger and Podolak's view of the original theorem. A description of folds in terms of properties of fibers gives new perspective to the usual hypotheses in the subject. The text is intended as a guide, outlining arguments and stating results which will be detailed elsewhere

Random Convex Hulls and Extreme Value Statistics

In this paper we study the statistical properties of convex hulls of $N$ random points in a plane chosen according to a given distribution. The points may be chosen independently or they may be correlated. After a non-exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem, we present a unifying approach, based on the notion of support function of a closed curve and the associated Cauchy's formulae, that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points. Our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics. As an example of correlated points, we study here in detail the case when the points represent the vertices of $n$ independent random walks. In the continuum time limit this reduces to $n$ independent planar Brownian trajectories for which we compute exactly, for all $n$, the mean perimeter and the mean area of their global convex hull. Our results have relevant applications in ecology in estimating the home range of a herd of animals. Some of these results were announced recently in a short communication [Phys. Rev. Lett. {\bf 103}, 140602 (2009)].Comment: 61 pages (pedagogical review); invited contribution to the special issue of J. Stat. Phys. celebrating the 50 years of Yeshiba/Rutgers meeting

An overview of the utilisation of microalgae biomass derived from nutrient recycling of wet market wastewater and slaughterhouse wastewater

Microalgae have high nutritional values for aquatic organisms compared to fish meal, because microalgae cells are rich in proteins, lipids, and carbohydrates. However, the high cost for the commercial production of microalgae biomass using fresh water or artificial media limits its use as fish feed. Few studies have investigated the potential of wet market wastewater and slaughterhouse wastewater for the production of microalgae biomass. Hence, this study aims to highlight the potential of these types of wastewater as an alternative superior medium for microalgae biomass as they contain high levels of nutrients required for microalgae growth. This paper focuses on the benefits of microalgae biomass produced during the phycore-mediation of wet market wastewater and slaughterhouse wastewater as fish feed. The extraction techniques for lipids and proteins as well as the studies conducted on the use of microalgae biomass as fish feed were reviewed. The results showed that microalgae biomass can be used as fish feed due to feed utilisation efficiency, physiological activity, increased resistance for several diseases, improved stress response, and improved protein retention

A local symmetry result for linear elliptic problems with solutions changing sign

We prove that the only domain Ω such that there exists a solution to the following problem in Ω, on ∂Ω, and , for a given constant c, is the unit ball , if we assume that Ω lies in an appropriate class of Lipschitz domains.Fil: Canuto, Bruno. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin

Multipulse phases in k-mixtures of Bose-Einstein condensates

For a competitive system of k coupled nonlinear Schroedinger equations we prove the existence, when the competition parameter is large, of positive radial solutions on R^N. We show that, when the competition parameter goes to infinity, the profile of each component separates, in many pulses, from the others. Moreover, we can prescribe the location of such pulses in terms of the oscillations of the changing-sign solutions of the scalar nonlinear Schroedinger equation. Within an Hartree-Fock approximation, this provides a theoretical indication of phase separation into many nodal domains for the k-mixtures of Bose-Einstein condensates.Comment: 21 page

The RESET project: constructing a European tephra lattice for refined synchronisation of environmental and archaeological events during the last c. 100 ka

This paper introduces the aims and scope of the RESET project (. RESponse of humans to abrupt Environmental Transitions), a programme of research funded by the Natural Environment Research Council (UK) between 2008 and 2013; it also provides the context and rationale for papers included in a special volume of Quaternary Science Reviews that report some of the project's findings. RESET examined the chronological and correlation methods employed to establish causal links between the timing of abrupt environmental transitions (AETs) on the one hand, and of human dispersal and development on the other, with a focus on the Middle and Upper Palaeolithic periods. The period of interest is the Last Glacial cycle and the early Holocene (c. 100-8 ka), during which time a number of pronounced AETs occurred. A long-running topic of debate is the degree to which human history in Europe and the Mediterranean region during the Palaeolithic was shaped by these AETs, but this has proved difficult to assess because of poor dating control. In an attempt to move the science forward, RESET examined the potential that tephra isochrons, and in particular non-visible ash layers (cryptotephras), might offer for synchronising palaeo-records with a greater degree of finesse. New tephrostratigraphical data generated by the project augment previously-established tephra frameworks for the region, and underpin a more evolved tephra 'lattice' that links palaeo-records between Greenland, the European mainland, sub-marine sequences in the Mediterranean and North Africa. The paper also outlines the significance of other contributions to this special volume: collectively, these illustrate how the lattice was constructed, how it links with cognate tephra research in Europe and elsewhere, and how the evidence of tephra isochrons is beginning to challenge long-held views about the impacts of environmental change on humans during the Palaeolithic. © 2015 Elsevier Ltd.RESET was funded through Consortium Grants awarded by the Natural Environment Research Council, UK, to a collaborating team drawn from four institutions: Royal Holloway University of London (grant reference NE/E015905/1), the Natural History Museum, London (NE/E015913/1), Oxford University (NE/E015670/1) and the University of Southampton, including the National Oceanography Centre (NE/01531X/1). The authors also wish to record their deep gratitude to four members of the scientific community who formed a consultative advisory panel during the lifetime of the RESET project: Professor Barbara Wohlfarth (Stockholm University), Professor Jørgen Peder Steffensen (Niels Bohr Institute, Copenhagen), Dr. Martin Street (Romisch-Germanisches Zentralmuseum, Neuwied) and Professor Clive Oppenheimer (Cambridge University). They provided excellent advice at key stages of the work, which we greatly valued. We also thank Jenny Kynaston (Geography Department, Royal Holloway) for construction of several of the figures in this paper, and Debbie Barrett (Elsevier) and Colin Murray Wallace (Editor-in-Chief, QSR) for their considerable assistance in the production of this special volume.Peer Reviewe