Boundary estimates in homogenization of systems of elasticity on Lipschitz domains: Lp theory

The primary purpose of this paper is to investigate a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients, arising in the theory of homogenization in Lipschitz domains. As a consequence, for $d\ge 4$, we prove that the $L^p$ Neumann and $L^p$ Dirichlet boundary value problems for systems of second order linear elasticity are uniquely solvable for $\frac{2(d?1)}{d+1}?\delta <p<2+\delta$ and $2?\delta <p<\frac{2(d?1)}{d?3}+\delta$ respectively.     

On the Hénon equation with a Neumann boundary condition: Asymptotic profile of ground states

Consider the Hénon equation with the homogeneous Neumann boundary condition

$?\mathrm{\Delta }u+u=|x{|}^{\alpha }{u}^p,u>0\text{in}\mathrm{\Omega },\frac{?u}{?\nu }=0\text{ on }?\mathrm{\Omega },$

where $\mathrm{\Omega }?B(0,1)?{\mathbb{R}}^N,N\ge 2$ and $?\mathrm{\Omega }\cap ?B(0,1)\ne ?$. We are concerned on the asymptotic behavior of ground state solutions as the parameter $\alpha \to \infty$. As $\alpha \to \infty$, the non-autonomous term $|x{|}^{\alpha }$ is getting singular near $|x|=1$. The singular behavior of $|x{|}^{\alpha }$ for large $\alpha >0$ forces the solution to blow up. Depending subtly on the $(N?1)?$dimensional measure $|?\mathrm{\Omega }\cap ?B(0,1){|}_{N?1}$ and the nonlinear growth rate p, there are many different types of limiting profiles. To catch the asymptotic profiles, we take different types of renormalization depending on p and $|?\mathrm{\Omega }\cap ?B(0,1){|}_{N?1}$. In particular, the critical exponent $2_{?}=\frac{2(N?1)}{N?2}$ for the Sobolev trace embedding plays a crucial role in the renormalization process. This is quite contrasted with the case of Dirichlet problems, where there is only one type of limiting profile for any $p\in (1,2^{?}?1)$ and a smooth domain Ω.     

Existence et unicité pour un fluide inhomogène

Recently R. Danchin showed the existence and uniqueness for an inhomogenous fluid in the homogeneous Besov space ${\dot{B}}_{21}^{\frac{N}{2}}({\mathbb{R}}^N)\times {\dot{B}}_{21}^{?1+\frac{N}{2}}({\mathbb{R}}^N)$, under the condition that ${\rho }_0?1$ is small in ${\dot{B}}_{2\infty }^{\frac{N}{2}}\cap L^{\infty }$ if $2<N,$ in ${\dot{B}}_{21}^{\frac{N}{2}}$ if $N=2.$ In this Note, one shows that the condition $6{\rho }_0?16_{L^{\infty }}?1$ is sufficient to have the existence and uniqueness. To cite this article: H. Abidi, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Sur les bornes inférieures de l'écart des variétés invariantes

Let m be an integer ?3, set $?(r)=\frac{1}{2}(r_1^2+?+r_{m?1}^2)+r_m$ for $r\in {\mathbb{R}}^m$, and consider a badly approximable vector ${\overline{\omega }}^0\in {\mathbb{R}}^{m?2}$. Fix $\alpha >1$, $L>0$ and $R>1+6{\overline{\omega }}^0{6}_{\infty }$. We construct a sequence $(H_N)$ of $\text{Gevrey}\text{-}(\alpha ,L)$ Hamiltonian functions of ${\mathbb{T}}^m\times {\overline{B}}_{\infty }(0,R)$, which converges to ? when $N\to \infty$, such that for each N the system generated by $H_N$ possesses a $(m?1)$-dimensional hyperbolic invariant torus with fixed frequency vector $({\overline{\omega }}^0,1)$, which admits a homoclinic point with splitting matrix of the form $\mathrm{diag}(0,{\nu }_N,\dots ,{\nu }_N,0)\in M_m(\mathbb{R})$, with ${\nu }_N?\exp (?c{(\frac{1}{{?}_N})}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2(\alpha ?1)(m?2)$}\right.})$, where ${?}_N:=6H_N??6_{\alpha ,L}$ and $c>0$. To cite this article: J.-P. Marco, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Solutions with interior bubble and boundary layer for an elliptic Neumann problem with critical nonlinearity

We study positive solutions of the equation ${\mathrm{?}}^2\mathrm{\Delta }u?u+u^{\frac{n+2}{n?2}}=0$, where $n=3,4,5$ and $\mathrm{?}>0$ is small, with Neumann boundary condition in a unit ball B. We prove the existence of solutions with an interior bubble at the center and a boundary layer at the boundary ?B. To cite this article: J. Wei, S. Yan, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

A characterization of a class of hyperplanes of DW(2n−1,F)

A hyperplane of the symplectic dual polar space $DW(2n?1,\mathbb{F})$, $n\ge 2$, is said to be of subspace-type if it consists of all maximal singular subspaces of $W(2n?1,\mathbb{F})$ meeting a given $(n?1)$-dimensional subspace of $\text{PG}(2n?1,\mathbb{F})$. We show that a hyperplane of $DW(2n?1,\mathbb{F})$ is of subspace-type if and only if every hex $F$ of $DW(2n?1,\mathbb{F})$ intersects it in either $F$, a singular hyperplane of $F$ or the extension of a full subgrid of a quad. In the case $\mathbb{F}$ is a perfect field of characteristic 2, a stronger result can be proved, namely a hyperplane $H$ of $DW(2n?1,\mathbb{F})$ is of subspace-type or arises from the spin-embedding of $DW(2n?1,\mathbb{F})?DQ(2n,\mathbb{F})$ if and only if every hex $F$ intersects it in either $F$, a singular hyperplane of $F$, a hexagonal hyperplane of $F$ or the extension of a full subgrid of a quad.     

Toughness,binding number and restricted matching extension in a graph

A connected graph $G$ with at least $2m+2n+2$ vertices is said to satisfy the property $E(m,n)$ if $G$ contains a perfect matching and for any two sets of independent edges $M$ and $N$ with $\left|M\right|=m$ and $\left|N\right|=n$ with $M\cap N=?$, there is a perfect matching $F$ in $G$ such that $M?F$ and $N\cap F=?$. In particular, if $G$ is $E(m,0)$, we say that $G$ is $m$-extendable. One of the authors has proved that every $m$-tough graph of even order at least $2m+2$ is $m$-extendable (Plummer, 1988). Chen (1995) and Robertshaw and Woodall (2002) gave sufficient conditions on binding number for $m$-extendability. In this paper, we extend these results and give lower bounds on toughness and binding number which guarantee $E(m,n)$.     

The NLC-width and clique-width for powers of graphs of bounded tree-width

The $k$-power graph of a graph $G$ is a graph with the same vertex set as $G$, in that two vertices are adjacent if and only if, there is a path between them in $G$ of length at most $k$. A $k$-tree-power graph is the $k$-power graph of a tree, a $k$-leaf-power graph is the subgraph of some $k$-tree-power graph induced by the leaves of the tree.We show that (1) every $k$-tree-power graph has NLC-width at most $k+2$ and clique-width at most $k+2+max\{?\frac{k}{2}??1,0\}$, (2) every $k$-leaf-power graph has NLC-width at most $k$ and clique-width at most $k+max\{?\frac{k}{2}??2,0\}$, and (3) every $k$-power graph of a graph of tree-width $l$ has NLC-width at most ${(k+1)}^{l+1}?1$, and clique-width at most $2?{(k+1)}^{l+1}?2$.     

Régularité conditionnelle des équations de Navier–Stokes

In this Note, we give sufficient conditions for the regularity of Leray–Hopf weak solutions to the Navier–Stokes equation. We prove that, if one of three conditions (i) $?u/?x_3\in L_t^{s_0}{L}_x^{r_0}$ where $2/s_0+3/r_0?2$ and $9/4?r_0?3$, (ii) $\mathrm{?}{u}_3\in L_t^{s_1}{L}_x^{r_1}$ where $2/s_1+3/r_1?11/6$ and $54/23?r_0?18/5$, or (iii) $u_3\in L_t^{s_0}{L}_x^{r_0}$ where $2/s_0+3/r_0?5/8$ and $24/5?r_0?\infty$, is satisfied, then the solution is regular. These conditions improve earlier results on the conditional regularity of the Navier–Stokes equations. To cite this article: I. Kukavica, M. Ziane, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

On packing of rectangles in a rectangle

It is known that ${\sum }_{i=1}^{\infty }1∕\left(i(i+1)\right)=1$. In 1968, Meir and Moser (1968) asked for finding the smallest $?$ such that all the rectangles of sizes $1∕i\times 1∕(i+1)$, $i\in \{1,2,\dots \}$, can be packed into a square or a rectangle of area $1+?$. First we show that in Paulhus (1997), the key lemma, as a statement, in the proof of the smallest published upper bound of the minimum area is false, then we prove a different new upper bound. We show that $?\le 1.26?10^{?9}$ if the rectangles are packed into a square and $?\le 6.878?10^{?10}$ if the rectangles are packed into a rectangle.