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 $2?{(k+1)}^{l+1}?2$.     

Optimal Wloc2,2-regularity,Pohozhaevʼs identity,and nonexistence of weak solutions to some quasilinear elliptic equations

We begin by establishing a sharp (optimal) $W_{\mathrm{loc}}^{2,2}$-regularity result for bounded weak solutions to a nonlinear elliptic equation with the p-Laplacian, ${\mathrm{\Delta }}_{p}u\stackrel{\mathrm{def}}{=}\mathrm{div}({|\mathrm{?}u|}^{p?2}\mathrm{?}u)$, $1<p<\infty$. We develop very precise, optimal regularity estimates on the ellipticity of this degenerate (for $2<p<\infty$) or singular (for $1<p<2$) problem. We apply this regularity result to prove Pohozhaev?s identity for a weak solution $u\in W^{1,p}(\Omega )$ of the elliptic Neumann problem(P)$?{\mathrm{\Delta }}_{p}u+W^{\prime }(u)=f(x)\text{in}\Omega ;?u/?\nu =0\text{on}?\Omega .$ Here, Ω is a bounded domain in ${\mathbb{R}}^N$ whose boundary ?Ω is a $C^2$-manifold, $\nu \equiv \nu (x_0)$ denotes the outer unit normal to ?Ω at $x_0\in ?\Omega$, $x=(x_1,\dots ,x_N)$ is a generic point in Ω, and $f\in L^{\infty }(\Omega )\cap W^{1,1}(\Omega )$. The potential $W:\mathbb{R}\to \mathbb{R}$ is assumed to be of class $C^1$ and of the typical double-well shape of type $W(s)={|1?{|s|}^{\beta }|}^{\alpha }$ for $s\in \mathbb{R}$, where $\alpha ,\beta >1$ are some constants. Finally, we take an advantage of the Pohozhaev identity to show that problem (P) with $f\equiv 0$ in Ω has no phase transition solution $u\in W^{1,p}(\Omega )$ ($1<p?N$), such that $?1?u?1$ in Ω with $u\equiv ?1$ in ${\Omega }_{?1}$ and $u\equiv 1$ in ${\Omega }_1$, where both ${\Omega }_{?1}$ and ${\Omega }_1$ are some nonempty subdomains of Ω. Such a scenario for u is possible only if $N=1$ and ${\Omega }_{?1}$, ${\Omega }_1$ are finite unions of suitable subintervals of the open interval $\Omega ?{\mathbb{R}}^1$.