Local C1 stability versus global C1 unstability for iterative roots

Stability of iterative roots is important in the numerical computation of iterative roots. Known results show that under some conditions iterative roots of strictly monotonic self-mappings are $C^0$ stable in both the local sense and the global sense. In this paper we discuss the $C^1$ stability for iterative roots of strictly increasing self-mappings on a compact interval between two fixed points. We prove that those iterative roots are locally $C^1$ stable but globally $C^1$ unstable.     

Sur l'équation fonctionnelle ∫T(ψ(t+s)−ψ(s))3ds=sint

Clearly the equation given in the title has no solution in $C^{\alpha }$, $\alpha >\frac{1}{3}$. We give an explicit solution in $C^{1/3}$. The motivation for this question, and other questions of the same type, comes from the forthcoming article by H. Brezis. To cite this article: J.-P. Kahane, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

On the nonexistence of parabolic boundary points of certain domains in C2

In this paper, the nonexistence of parabolic boundary points of infinite type of certain domains in ${\mathbb{C}}^2$ is showed.     

Lower bounds for blow-up in a model of chemotaxis

Many special cases of the classical Keller–Segel system for modeling chemotaxis have been investigated in the literature, and typically the solution of the governing equations will blow up at some finite time. However, the question of establishing lower bounds for this blow-up time has been largely ignored. This paper derives such a lower bound in a parabolic–parabolic model in both ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$.     

Closed hypersurfaces of S4(1) with constant mean curvature and zero Gauß–Kronecker curvature

We consider a closed hypersurface $M^3?{\mathbb{S}}^4(1)$ with identically zero Gauß–Kronecker curvature. We prove that if $M^3$ has constant mean curvature H, then $M^3$ is minimal, i.e., $H=0$. This result extends Ramanathan's classification (Math. Z. 205 (1990) 645–658) result of closed minimal hypersurfaces of ${\mathbb{S}}^4(1)$ with vanishing Gauß–Kronecker curvature. To cite this article: T. Lusala, A. Gomes de Oliveira, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Inverse spectral problem for singular AKNS and Schrödinger operators on [0,1]

We consider an inverse spectral problem for singular Sturm–Liouville equations on the unit interval with explicit singularity $a(a+1)/x^2$, $a\in \mathbb{N}$. This problem arises by splitting of the Schrödinger operator with radial potential acting on the unit ball of ${\mathbb{R}}^3$. Our goal is the global parametrization of potentials by spectral data noted by ${\lambda }^a$, and some norming constants noted by ${\kappa }^a$. For $a=0$ and 1, ${\lambda }^a\times {\kappa }^a$ was already known to be a global coordinate system on $L_{\mathbb{R}}^2(0,1)$. We extend this to any non-negative integer a. Similar result is obtained for singular AKNS operator. To cite this article: F. Serier, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

A1-homotopy groups,excision, and solvable quotients

We study some properties of ${\mathbb{A}}^1$-homotopy groups: geometric interpretations of connectivity, excision results, and a re-interpretation of quotients by free actions of connected solvable groups in terms of covering spaces in the sense of ${\mathbb{A}}^1$-homotopy theory. These concepts and results are well suited to the study of certain quotients via geometric invariant theory. As a case study in the geometry of solvable group quotients, we investigate ${\mathbb{A}}^1$-homotopy groups of smooth toric varieties. We give simple combinatorial conditions (in terms of fans) guaranteeing vanishing of low degree ${\mathbb{A}}^1$-homotopy groups of smooth (proper) toric varieties. Finally, in certain cases, we can actually compute the “next” non-vanishing ${\mathbb{A}}^1$-homotopy group (beyond ${\pi }_1^{{\mathbb{A}}^1}$) of a smooth toric variety. From this point of view, ${\mathbb{A}}^1$-homotopy theory, even with its exquisite sensitivity to algebro-geometric structure, is almost “as tractable” (in low degrees) as ordinary homotopy for large classes of interesting varieties.     

Convergence rates and interior estimates in homogenization of higher order elliptic systems

This paper is concerned with the quantitative homogenization of 2m-order elliptic systems with bounded measurable, rapidly oscillating periodic coefficients. We establish the sharp $O(\epsilon )$ convergence rate in $W^{m?1,p_0}$ with $p_0=\frac{2d}{d?1}$ in a bounded Lipschitz domain in ${\mathbb{R}}^d$ as well as the uniform large-scale interior $C^{m?1,1}$ estimate. With additional smoothness assumptions, the uniform interior $C^{m?1,1}$, $W^{m,p}$ and $C^{m?1,\alpha }$ estimates are also obtained. As applications of the regularity estimates, we establish asymptotic expansions for fundamental solutions.     

Critical space for the parabolic-parabolic Keller–Segel model in Rd

We study the Keller–Segel system in ${\mathbb{R}}^d$ when the chemoattractant concentration is described by a parabolic equation. We prove that the critical space, with some similarity to the elliptic case, is that the initial bacteria density satisfies $n_0\in L^a({\mathbb{R}}^d)$, $a>d/2$, and that the chemoattractant concentration satisfies $\mathrm{?}{c}_0\in L^d({\mathbb{R}}^d)$. In these spaces, we prove that small initial data give rise to global solutions that vanish as the heat equation for large times and that exhibit a regularizing effect of hypercontractivity type. To cite this article: L. Corrias, B. Perthame, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Capacitary representation of positive solutions of semilinear parabolic equations

We give a global bilateral estimate on the maximal solution ${\overline{u}}_F$ of ${?}_{t}u?\mathrm{\Delta }u+u^q=0$ in ${\mathbb{R}}^N\times (0,\infty )$, $q>1$, $N?1$, which vanishes at $t=0$ on the complement of a closed subset $F?{\mathbb{R}}^N$. This estimate is expressed by a Wiener test involving the Bessel capacity $C_{2/q,q^{\prime }}$. We deduce from this estimate that ${\overline{u}}_F$ is σ-moderate in Dynkin's sense. To cite this article: M. Marcus, L. Véron, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Théorème de Liouville et propriété de la moyenne biharmonique restreinte

We prove, under some conditions, that a bounded Lebesgue measurable function satisfying the restricted mean value for the biharmonic functions in ${\mathbf{R}}^n$, $n\ne 2$, or in an open set of ${\mathbf{R}}^2$ with polar complement, is constant. To cite this article: M. El Kadiri, C. R. Acad. Sci. Paris, Ser. I 340 (2005).