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

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

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

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

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

Initial Upper Palaeolithic humans in Europe had recent Neanderthal ancestry

Modern humans appeared in Europe by at least 45,000 years ago1–5, but the extent of their interactions with Neanderthals, who disappeared by about 40,000 years ago6, and their relationship to the broader expansion of modern humans outside Africa are poorly understood. Here we present genome-wide data from three individuals dated to between 45,930 and 42,580 years ago from Bacho Kiro Cave, Bulgaria1,2. They are the earliest Late Pleistocene modern humans known to have been recovered in Europe so far, and were found in association with an Initial Upper Palaeolithic artefact assemblage. Unlike two previously studied individuals of similar ages from Romania7 and Siberia8 who did not contribute detectably to later populations, these individuals are more closely related to present-day and ancient populations in East Asia and the Americas than to later west Eurasian populations. This indicates that they belonged to a modern human migration into Europe that was not previously known from the genetic record, and provides evidence that there was at least some continuity between the earliest modern humans in Europe and later people in Eurasia. Moreover, we find that all three individuals had Neanderthal ancestors a few generations back in their family history, confirming that the first European modern humans mixed with Neanderthals and suggesting that such mixing could have been common

Multi-Level Interactions between the Nuclear Receptor TRα1 and the WNT Effectors β-Catenin/Tcf4 in the Intestinal Epithelium

Intestinal homeostasis results from complex cross-regulation of signaling pathways; their alteration induces intestinal tumorigenesis. Previously, we found that the thyroid hormone nuclear receptor TRα1 activates and synergizes with the WNT pathway, inducing crypt cell proliferation and promoting tumorigenesis. Here, we investigated the mechanisms and implications of the cross-regulation between these two pathways in gut tumorigenesis in vivo and in vitro. We analyzed TRα1 and WNT target gene expression in healthy mucosae and tumors from mice overexpressing TRα1 in the intestinal epithelium in a WNT-activated genetic background (vil-TRα1/Apc mice). Interestingly, increased levels of β-catenin/Tcf4 complex in tumors from vil-TRα1/Apc mice blocked TRα1 transcriptional activity. This observation was confirmed in Caco2 cells, in which TRα1 functionality on a luciferase reporter-assay was reduced by the overexpression of β-catenin/Tcf4. Moreover, TRα1 physically interacted with β-catenin/Tcf4 in the nuclei of these cells. Using molecular approaches, we demonstrated that the binding of TRα1 to its DNA target sequences within the tumors was impaired, while it was newly recruited to WNT target genes. In conclusion, our observations strongly suggest that increased β-catenin/Tcf4 levels i) correlated with reduced TRα1 transcriptional activity on its target genes and, ii) were likely responsible for the shift of TRα1 binding on WNT targets. Together, these data suggest a novel mechanism for the tumor-promoting activity of the TRα1 nuclear receptor