Asymptotic behaviour of a semilinear elliptic system with a large exponent

Consider the problem \begin{eqnarray*} -\Delta u &=& v^{\frac 2{N-2}},\quad v>0\quad {in}\quad \Omega, -\Delta v &=& u^{p},\:\:\:\quad u>0\quad {in}\quad \Omega, u&=&v\:\:=\:\:0 \quad {on}\quad \partial \Omega, \end{eqnarray*} where $\Omega$ is a bounded convex domain in $\R^N,$ $N>2,$ with smooth boundary $\partial \Omega.$ We study the asymptotic behaviour of the least energy solutions of this system as $p\to \infty.$ We show that the solution remain bounded for $p$ large and have one or two peaks away form the boundary. When one peak occurs we characterize its location.Comment: 16 pages, submmited for publicatio

Moment inversion problem for piecewise D-finite functions

We consider the problem of exact reconstruction of univariate functions with jump discontinuities at unknown positions from their moments. These functions are assumed to satisfy an a priori unknown linear homogeneous differential equation with polynomial coefficients on each continuity interval. Therefore, they may be specified by a finite amount of information. This reconstruction problem has practical importance in Signal Processing and other applications. It is somewhat of a ``folklore'' that the sequence of the moments of such ``piecewise D-finite''functions satisfies a linear recurrence relation of bounded order and degree. We derive this recurrence relation explicitly. It turns out that the coefficients of the differential operator which annihilates every piece of the function, as well as the locations of the discontinuities, appear in this recurrence in a precisely controlled manner. This leads to the formulation of a generic algorithm for reconstructing a piecewise D-finite function from its moments. We investigate the conditions for solvability of the resulting linear systems in the general case, as well as analyze a few particular examples. We provide results of numerical simulations for several types of signals, which test the sensitivity of the proposed algorithm to noise

The dynamics of the 3D radial NLS with the combined terms

In this paper, we show the scattering and blow-up result of the radial solution with the energy below the threshold for the nonlinear Schr\"{o}dinger equation (NLS) with the combined terms iu_t + \Delta u = -|u|^4u + |u|^2u \tag{CNLS} in the energy space $H^1(\R^3)$. The threshold is given by the ground state $W$ for the energy-critical NLS: $iu_t + \Delta u = -|u|^4u$. This problem was proposed by Tao, Visan and Zhang in \cite{TaoVZ:NLS:combined}. The main difficulty is the lack of the scaling invariance. Illuminated by \cite{IbrMN:f:NLKG}, we need give the new radial profile decomposition with the scaling parameter, then apply it into the scattering theory. Our result shows that the defocusing, $\dot H^1$-subcritical perturbation $|u|^2u$ does not affect the determination of the threshold of the scattering solution of (CNLS) in the energy space.Comment: 46page

Second-order L2L^2-regularity in nonlinear elliptic problems

A second-order regularity theory is developed for solutions to a class of quasilinear elliptic equations in divergence form, including the $p$-Laplace equation, with merely square-integrable right-hand side. Our results amount to the existence and square integrability of the weak derivatives of the nonlinear expression of the gradient under the divergence operator. This provides a nonlinear counterpart of the classical $L^2$-coercivity theory for linear problems, which is missing in the existing literature. Both local and global estimates are established. The latter apply to solutions to either Dirichlet or Neumann boundary value problems. Minimal regularity on the boundary of the domain is required. If the domain is convex, no regularity of its boundary is needed at all

Genome-wide SNP profiling of worldwide goat populations reveals strong partitioning of diversity and highlights post-domestication migration routes

Background: Goat populations that are characterized within the AdaptMap project cover a large part of the worldwide distribution of this species and provide the opportunity to assess their diversity at a global scale. We analysed genome-wide 50 K single nucleotide polymorphism (SNP) data from 144 populations to describe the global patterns of molecular variation, compare them to those observed in other livestock species, and identify the drivers that led to the current distribution of goats. Results: A high degree of genetic variability exists among the goat populations studied. Our results highlight a strong partitioning of molecular diversity between and within continents. Three major gene pools correspond to goats from Europe, Africa and West Asia. Dissection of sub-structures disclosed regional gene pools, which reflect the main post-domestication migration routes. We also identified several exchanges, mainly in African populations, and which often involve admixed and cosmopolitan breeds. Extensive gene flow has taken place within specific areas (e.g., south Europe, Morocco and Mali-Burkina Faso-Nigeria), whereas elsewhere isolation due to geographical barriers (e.g., seas or mountains) or human management has decreased local gene flows. Conclusions: After domestication in the Fertile Crescent in the early Neolithic era (ca. 12,000 YBP), domestic goats that already carried differentiated gene pools spread to Europe, Africa and Asia. The spread of these populations determined the major genomic background of the continental populations, which currently have a more marked subdivision than that observed in other ruminant livestock species. Subsequently, further diversification occurred at the regional level due to geographical and reproductive isolation, which was accompanied by additional migrations and/or importations, the traces of which are still detectable today. The effects of breed formation were clearly detected, particularly in Central and North Europe. Overall, our results highlight a remarkable diversity that occurs at the global scale and is locally partitioned and often affected by introgression from cosmopolitan breeds. These findings support the importance of long-term preservation of goat diversity, and provide a useful framework for investigating adaptive introgression, directing genetic improvement and choosing breeding targets

Orbital stability of spherical galactic models

International audienceWe consider the three dimensional gravitational Vlasov Poisson system which is a canonical model in astrophysics to describe the dynamics of galactic clusters. A well known conjecture is the stability of spherical models which are nonincreasing radially symmetric steady states solutions. This conjecture was proved at the linear level by several authors in the continuation of the breakthrough work by Antonov in 1961. In a previous work, we derived the stability of anisotropic models under {\it spherically symmetric perturbations} using fundamental monotonicity properties of the Hamiltonian under suitable generalized symmetric rearrangements first observed in the physics litterature. In this work, we show how this approach combined with a {\it new generalized} Antonov type coercivity property implies the orbital stability of spherical models under general perturbations

A cattle graph genome incorporating global breed diversity

Despite only 8% of cattle being found in Europe, European breeds dominate current genetic resources. This adversely impacts cattle research in other important global cattle breeds, especially those from Africa for which genomic resources are particularly limited, despite their disproportionate importance to the continent's economies. To mitigate this issue, we have generated assemblies of African breeds, which have been integrated with genomic data for 294 diverse cattle into a graph genome that incorporates global cattle diversity. We illustrate how this more representative reference assembly contains an extra 116.1 Mb (4.2%) of sequence absent from the current Hereford sequence and consequently inaccessible to current studies. We further demonstrate how using this graph genome increases read mapping rates, reduces allelic biases and improves the agreement of structural variant calling with independent optical mapping data. Consequently, we present an improved, more representative, reference assembly that will improve global cattle research