Random Euclidean embeddings in spaces of bounded volume ratio

Let $\text{(}\text{R}{}^{\mathrm{N}}\text{,6·6)}$ be the space $\text{R}{}^{\mathrm{N}}$ equipped with a norm 6·6 whose unit ball has a bounded volume ratio with respect to the Euclidean unit ball. Let Γ be any random N×n matrix with N>n, whose entries are independent random variables satisfying some moment assumptions. We show that with high probability Γ is a good isomorphism from the n-dimensional Euclidean space $\text{(}\text{R}{}^{\mathrm{n}}\text{,|·|)}$ onto its image in $\text{(}\text{R}{}^{\mathrm{N}}\text{,6·6)}$: there exist α,β>0 such that for all $\text{x∈}\text{R}{}^{\mathrm{n}}$, $\text{α}\text{N}\text{|x|?6Γx6?β}\text{N}\text{|x|}$. This solves a conjecture of Schechtman on random embeddings of ?₂ⁿ into ?₁^N. To cite this article: A. Litvak et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Valeurs propres relatives à un cône convexe : caractérisation et résultats de cardinalité

Let A be an n×n real matrix, and $\text{K?}\text{R}{}^{\mathrm{n}}$ be a closed convex cone. The spectrum of A relative to K, denoted by σ(A,K), is the set of all $\text{λ∈}\text{R}$ for which the linear complementarity problem

$\text{x∈K,}\text{Ax?λx∈K}{}^{\mathrm{+}}\text{,}\text{〈x,Ax?λx〉=0}$

admits a nonzero solution $\text{x∈}\text{R}{}^{\mathrm{n}}$. The aim of this Note is to study the main properties of the set-valued mapping $\text{σ(·}\text{,K)}$, and discuss some structural differences existing between the polyhedral case (i.e., K is finitely generated) and the non-polyhedral case. To cite this article: A. Seeger, M. Torki, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On Lp estimates for the wave equation

New and more elementary proofs are given of two results due to W. Littman: (1) Let $\text{n ? 2, p ?}\text{2n}\text{(n ? 1)}$. The estimate $\text{∫∫ (¦▽u¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{¦}{}^{\mathrm{p}}\text{) dx dt ? C ∫∫ ¦□u¦}{}^{\mathrm{p}}\text{dx dt}$ cannot hold for all u?C₀^∞(Q), Q a cube in $\text{R}{}^{\mathrm{<mi\; mathvariant="italic"\; is="true">n</mi>}}\text{× R}$, some constant C. (2) Let n ? 2, p ≠ 2. The estimate $\text{∫ (¦▽(t)¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{(t)¦}{}^{\mathrm{p}}\text{) dx ? C(t) ∫ (¦▽u(0)¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{(0)¦}{}^{\mathrm{p}}\text{) dx}$ cannot hold for all C^∞ solutions of the wave equation □u = 0 in $\text{R}{}^{\mathrm{n}}\text{x R}$; all t ?$\text{R}$; some function C: $\text{R}$ → $\text{R}$.     

A variant of Poincaré's inequality

We show that if $\text{Ω?}\text{R}{}^{\mathrm{N}}\text{,}\text{N?2}$, is a bounded Lipschitz domain and $\text{(ρ}{}_{\mathrm{n}}\text{)?L}{}^1\text{(}\text{R}{}^{\mathrm{N}}\text{)}$ is a sequence of nonnegative radial functions weakly converging to δ₀ then there exist C>0 and n₀?1 such that

$\text{∫}\text{Ω}\text{f??}\text{∫}\text{Ω}\text{f}{}^{\mathrm{p}}\text{?C}\text{∫}\text{Ω}\text{∫}\text{Ω}\text{|f(x)?f(y)|}{}^{\mathrm{p}}\text{|x?y|}{}^{\mathrm{p}}\text{ρ}{}_{\mathrm{n}}\text{(|x?y|)}\text{d}\text{x}\text{d}\text{y}\text{?f∈L}{}^{\mathrm{p}}\text{(Ω)}\text{?n?n}{}_0\text{.}$

The above estimate was suggested by some recent work of Bourgain, Brezis and Mironescu (in: Optimal Control and Partial Differential Equations, IOS Press, 2001, pp. 439–455). As n→∞ in (1) we recover Poincaré's inequality. We also extend a compactness result of Bourgain, Brezis and Mironescu. To cite this article: A.C. Ponce, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Extension of a Riemannian metric with vanishing curvature

Let $\text{Ω}$ be a connected and simply-connected open subset of $\text{R}{}^{\mathrm{n}}$ such that the geodesic distance in $\text{Ω}$ is equivalent to the Euclidean distance. Let there be given a Riemannian metric (g_ij) of class $\text{C}{}^2$ and of vanishing curvature in $\text{Ω}$, such that the functions g_ij and their partial derivatives of order $\text{?}\text{2}$ have continuous extensions to $\text{Ω}$. Then there exists a connected open subset $\text{Ω}$ of $\text{R}{}^{\mathrm{n}}$ containing $\text{Ω}$ and a Riemannian metric $\text{(}\text{g}\text{?}{}_{\mathrm{ij}}\text{)}$ of class $\text{C}{}^2$ and of vanishing curvature in $\text{Ω}$ that extends the metric (g_ij). To cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Energy concentration and Sommerfeld condition for Helmholtz and Liouville equations

We consider the Helmholtz equation with a variable index of refraction n(x), which is not necessarily constant at infinity but can have an angular dependency like n(x)→n_∞(x/|x|) as |x|→∞. We prove that the Sommerfeld condition at infinity still holds true under the weaker form

$\text{1}\text{R}\text{∫}\text{|x|?R}\text{?u?}\text{i}\text{n}{}_{\mathrm{\infty }}{}^{\mathrm{1/2}}\text{x}\text{|x|}\text{u}\text{x}\text{|x|}{}^2\text{d}\text{x→0,}\text{as}\text{R→∞.}$

Our approach consists in proving this estimate in the framework of the limiting absorbtion principle. We use Morrey–Campanato type of estimates and a new inequality on the energy decay, namely

$\text{∫}\text{R}{}^{\mathrm{d}}\text{?}\text{?ω}\text{n}{}_{\mathrm{\infty }}\text{(ω)}{}^2\text{|u|}{}^2\text{|x|}\text{d}\text{x?C,}\text{ω=}\text{x}\text{|x|}\text{.}$

It is a striking feature that the index n_∞ appears in this formula and not the phase gradient, in apparent contradiction with existing literature. To cite this article: B. Perthame, L. Vega, C. R. Acad. Sci. Paris, Ser. I 337 (2003).