Exposants de Lyapounov pour opérateurs de Schrödinger discrètes quasi-périodiquesLyapounov exponents for discrete,quasi-periodic Schrödinger operators

We consider quasi-periodic Schrödinger operators H on $\text{Z}$ of the form H=H_λ,x,ω=λv(x+nω)δ_n,n′+Δ where v is a non-constant real analytic function on the d-torus $\text{T}{}^{\mathrm{d}}\text{(d?1)}$ and Δ denotes the discrete lattice Laplacian on $\text{Z}$. Denote by L_ω(E) the Lyapounov exponent, considered as function of the energy E and the rotation vector $\text{ω∈}\text{T}{}^{\mathrm{d}}$. It is shown that for |λ|>λ₀(v), there is the uniform minoration $\text{L}{}_{\mathrm{\omega }}\text{(E)>}\text{1}\text{2}\text{log}\text{|λ|}$ for all E and ω. For all λ and ω, L_ω(E) is a continuous function of E. Moreover, L_ω(E) is jointly continuous in (ω,E), at any point $\text{(ω}{}_0\text{,E}{}_0\text{)∈}\text{T}{}^{\mathrm{d}}\text{×}\text{R}$ such that k·ω₀≠0 for all $\text{k∈}\text{Z}{}^{\mathrm{d}}\text{?{0}}$. To cite this article: J. Bourgain, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 529–531.     

Propriétés de l'intégrale de Cauchy Harish-Chandra pour certaines paires duales d'algèbres de LieProperties of the Cauchy Harish-Chandra integral for some dual pairs of Lie algebras

We consider a symplectic group Sp and an reductive and irreductible dual pair (G,G′) in Sp in the sense of R. Howe. Let $\text{g}$ (resp. $\text{g}\text{′}$) be the Lie algebra of G (resp. G′). T. Przebinda has defined a map $\text{Chc}$, called the Cauchy Harish-Chandra integral from the space of smooth compactly supported functions of $\text{g}$ to the space of functions defined on the open set $\text{g}{}^{\mathrm{<mtext>\prime </mtext><mspace\; sp="0.2"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>reg</mtext>}}$ of semisimple regular elements of $\text{g}\text{′}$. We prove that these functions are invariant integrals if G and G′ are linear groups and they behave locally like invariant integrals if G and G′ are unitary groups of same rank. In this last case, we obtain the jump relations up to a multiplicative constant which only depends on the dual pair. To cite this article: F. Bernon, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 945–948.     

Sur quelques résultats de convergence dans l'inférence d'une classe de processus ponctuelsConvergence results for point processes estimators

Our aim is to generalize some results obtained for a Poisson point process in [7], to a general point process. Those results are in field of complete convergence of two like Parzen–Rosenblatt estimates of density of mean measure function and regression curves. Those estimates are defined from the superposition of n i.i.d. point processes as:

$\text{f}{}_{\mathrm{n}}\text{(x)=}\text{1}\text{nh}\text{∑}\text{i=1}\text{m}\text{K}\text{x?X}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{h(n)}\text{and}\text{Ψ}{}_{\mathrm{n}}\text{(x)=}\text{∑}{}_{\mathrm{i=1}}{}^{\mathrm{m}}\text{Y}{}_{\mathrm{i}}\text{K}\text{x?X}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{h(n)}\text{∑}{}_{\mathrm{i=1}}{}^{\mathrm{m}}\text{K}\text{x?X}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{h(n)}\text{,}$

where m is the number of seem generics points of the superposition. We give some sufficient conditions for the convergence of those kernel-like estimators. To cite this article: A. Diakhaby, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 597–602.     

Sharp Sobolev type inequalities for higher fractional derivativesInégalités optimales de type Sobolev pour les dérivées fractionelles d'ordre supérieur

On $\text{R}{}^{\mathrm{n}}$, n?1 and n≠2, we prove the existence of a sharp constant for Sobolev inequalities with higher fractional derivatives. Let s be a positive real number. For n>2s and $\text{q=}\text{2n}\text{n?2s}$ any function $\text{f∈}\text{H}{}^{\mathrm{s}}\text{(}\text{R}{}^{\mathrm{n}}\text{)}$ satisfies

$\text{6f6}{}^2{}_{\mathrm{q}}\text{?S}{}_{\mathrm{n,s}}\text{(?Δ)}{}^{\mathrm{s/2}}\text{f}{}^2{}_2\text{,}$

where the operator (?Δ)^s in Fourier spaces is defined by $\text{(?Δ)}{}^{\mathrm{s}}\text{f}\text{(k):=(2π|k|)}{}^{\mathrm{2s}}\text{f}\text{(k)}$. To cite this article: A. Cotsiolis, N.C. Tavoularis, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 801–804.     

On the “prediction” problemSur le problème de prédiction

We prove that almost every (in the Baire category sense) weight w on a circle $\text{T}$ satisfies the following property: any function from $\text{L}{}^2\text{(w,}\text{T}\text{)}$ can be decomposed as a series

$\text{∑}\text{n∈}\text{Z}{}^{\mathrm{+}}\text{c(n)}\text{e}{}^{\mathrm{int}}$

which converges in the norm.We discuss this result in the context of the classical Szegö–Kolmogorov “prediction” theorem. To cite this article: A. Olveskii, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 279–282.     

A sharp inequality for Sobolev functionsUne inégalité dans un espace de Sobolev

Let N?5, a>0, $\text{Ω}$ be a smooth bounded domain in $\text{R}{}^{\mathrm{N}}$, $\text{2}{}^1\text{=}\text{2N}\text{N?2}$, $\text{2}{}^{\mathrm{\#}}\text{=}\text{2(N?1)}\text{N?2}$ and 6u6²=|?u|₂²+a|u|₂². We prove there exists an α₀>0 such that, for all $\text{u∈}\text{H}{}^1\text{(Ω)?{0}}$,

This inequality implies Cherrier's inequality. To cite this article: P.M. Girão, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 105–108     

A comparison result related to Gauss measureUn résultat de comparaison relatif à la mesure de Gauss

In this paper we prove a comparison result for weak solutions to linear elliptic problems of the type

where $\text{Ω}$ is an open set of $\text{R}{}^{\mathrm{n}}$ (n?2), ?(x)=(2π)^?n/2exp(?|x|²/2), a_ij(x) are measurable functions such that a_ij(x)ξ_iξ_j??(x)|ξ|² a.e. $\text{x∈Ω}$, $\text{ξ∈}\text{R}{}^{\mathrm{n}}$ and f(x) is a measurable function taken in order to guarantee the existence of a solution $\text{u∈}\text{H}{}^1{}_0\text{(?,Ω)}$ of (1.1). We use the notion of rearrangement related to Gauss measure to compare u(x) with the solution of a problem of the same type, whose data are defined in a half-space and depend only on one variable. To cite this article: M.F. Betta et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 451–456.     

Nonlinear “double porosity” type modelUn modèle non linéaire de type double porosité

We consider a variational problem in a bounded domain $\text{Ω=}\text{F}{}^{\mathrm{(\epsilon )}}\text{∪}\text{M}{}^{\mathrm{(\epsilon )}}$ with a microstructure $\text{F}{}^{\mathrm{(\epsilon )}}$ which is not in general periodic; a^ε=a^ε(x) is of order 1 in $\text{F}{}^{\mathrm{(\epsilon )}}$ and as ε→0. A homogenized model is constructed. To cite this article: L. Pankratov, A. Piatnitski, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 435–440.     

Sur l'équation divu=fOn the equation divu=f

The main result is the following. Let $\text{Ω}$ be a bounded Lipschitz domain in $\text{R}{}^{\mathrm{d}}$, d?2. Then for every $\text{f∈}\text{L}{}^{\mathrm{d}}\text{(Ω)}$ with ∫f=0, there exists a solution $\text{u∈}\text{C}{}^0\text{(}\text{Ω}\text{)∩}\text{W}{}^{\mathrm{1,d}}\text{(Ω)}$ of the equation divu=f in $\text{Ω}$, satisfying in addition u=0 on $\text{?Ω}$ and the estimate

where C depends only on $\text{Ω}$. However one cannot choose u depending linearly on f. To cite this article: J. Bourgain, H. Brezis, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 973–976.     

Viscoélastodynamique monodimensionnelle avec conditions de SignoriniOne-dimensional viscoelastodynamics with Signorini boundary conditions

Let α be a positive number. The one-dimensional viscoelastic problem

$\text{u}{}_{\mathrm{tt}}\text{?u}{}_{\mathrm{xx}}\text{?αu}{}_{\mathrm{xxt}}\text{=f,}\text{x∈(?∞,0],}\text{t∈[0,+∞),}$

with unilateral boundary conditions

$\text{u(0,·)?0,}\text{(u}{}_{\mathrm{x}}\text{+αu}{}_{\mathrm{xt}}\text{)(0,·)?0,}\text{(u(u}{}_{\mathrm{x}}\text{+αu}{}_{\mathrm{xt}}\text{))(0,·)=0,}$

can be reduced to the following variational inequality:

$\text{λ}{}_1\text{1w=g+b,}\text{w?0,}\text{b?0,}\text{〈w,b〉=0.}$

Here $\text{λ}\text{?}{}_1\text{(ω)}$ is the causal determination of $\text{i}\text{ω}\text{1+}\text{i}\text{αω}$. We show that the energy losses are purely viscous; this result is a consequence of the relation $\text{〈}\text{w}\text{?}\text{,b〉=0}$; since a priori, b is a measure and $\text{w}\text{?}$ is defined only almost everywhere, this relation is not trivial. To cite this article: A. Petrov, M. Schatzman, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 983–988.     

Propagation of chaos for pressureless gas equations with viscosityPropagation du chaos pour un système de gaz sans pression avec viscosité

We use A.S. Sznitman ideas of probabilistic phenomenon of propagation of chaos for Burgers equation, and we derive the existence and uniqueness of a weak solution of the following system of pressureless gas equations with viscosity:

$\text{(}\text{S}\text{)}\text{?}\text{?t}\text{ρ+}\text{?}\text{?x}\text{(uρ)=}\text{1}\text{2}\text{?}{}^2\text{?}{}^2\text{x}\text{ρ,}\text{?}\text{?t}\text{(uρ)+}\text{?}\text{?x}\text{(u}{}^2\text{ρ)=}\text{1}\text{2}\text{?}{}^2\text{?}{}^2\text{x}\text{(uρ),}\text{ρ(}\text{d}\text{x,t)→ρ(}\text{d}\text{x,0),u(x,t)ρ(}\text{d}\text{x,t)→u}{}_0\text{(x)ρ(}\text{d}\text{x,0)}\text{weakly as}\text{t→0}{}^{\mathrm{+}}\text{.}$

To cite this article: A. Dermoune, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 935–940.     

Asymptotic properties of posterior distributions derived from misspecified modelsPropriétés asymptotiques des lois a posteriori sous un modèle incorrect

We investigate the asymptotic properties of posterior distributions when the model is misspecified, i.e. it is contemplated that the observations x₁,…,x_n might be drawn from a density in a family $\text{{h}{}_{\mathrm{\sigma }}\text{,}\text{σ∈Θ}}$ where $\text{Θ?}\text{R}{}^{\mathrm{d}}$, while the actual distribution of the observations may not correspond to any of the densities h_σ. A concentration property around a fixed value of the parameter is obtained as well as concentration properties around the maximum likelihood estimate. To cite this article: C. Abraham, B. Cadre, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 495–498.     

On the asymptotics of global solutions of higher-order semilinear parabolic equations in the supercritical rangeSur les asymptotiques des solutions globales des équations paraboliques sémi-linéaires d'ordre supérieur dans le cas surcritique

We study the asymptotic behaviour of global bounded solutions of the Cauchy problem for the semilinear 2mth order parabolic equation u_t=?(?Δ)^mu+|u|^p in R^N×R₊, where m>1, p>1, with bounded integrable initial data u₀. We prove that in the supercritical Fujita range p>p_F=1+2m/N any small global solution with nonnegative initial mass, $\text{∫u}{}_0\text{dx?0}$, exhibits as t→∞ the asymptotic behaviour given by the fundamental solution of the linear parabolic operator (unlike the case $\text{p∈}\text{]1,p}{}_{\mathrm{F}}\text{]}$ where solutions can blow-up for any arbitrarily small initial data). A discrete spectrum of other possible asymptotic patterns and the corresponding monotone sequence of critical exponents $\text{{p}{}_{\mathrm{l}}\text{=1+2m/(l+N),}\text{l=0,1,2,…}}$, where p₀=p_F, are discussed. To cite this article: Yu.V. Egorov et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 805–810.     

On global discontinuous solutions of Hamilton–Jacobi equationsSur des solutions globales discontinues des équations d'Hamilton–Jacobi

The uniqueness of classical semicontinuous viscosity solutions of the Cauchy problem for Hamilton–Jacobi equations is established for globally Lipschitz continuous and convex Hamiltonian H=H(Du), provided the discontinuous initial value function ?(x) is continuous outside a set Γ of measure zero and satisfies

(1)$\text{?(x)??}{}_7\text{(x):=}\text{lim}\text{inf}\text{y→x,}\text{y∈}\text{R}{}^{\mathrm{d}}\text{?Γ}\text{?(y).}$

We prove that the discontinuous solutions with almost everywhere continuous initial data satisfying ($\text{1}$) become Lipschitz continuous after finite time for locally strictly convex Hamiltonians. The L¹-accessibility of initial data and a comparison principle for discontinuous solutions are shown for a general Hamiltonian. The equivalence of semicontinuous viscosity solutions, bi-lateral solutions, L-solutions, minimax solutions, and L^∞-solutions is clarified. To cite this article: G.-Q. Chen, B. Su, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 113–118     

Construction and stability of blowup solutions for a non-variational semilinear parabolic systemConstruction et stabilité de solutions explosives pour un système parabolique sémilinéaire non-variationel

We consider the following parabolic system whose nonlinearity has no gradient structure:

$\{\begin{array}{cc}{?}_{t}u=\mathrm{\Delta }u+|v{|}^{p?1}v,\hfill & {?}_{t}v=\mu \mathrm{\Delta }v+|u{|}^{q?1}u,\hfill \\ u(?,0)=u_0,\hfill & v(?,0)=v_0,\hfill \end{array}$

in the whole space ${\mathbb{R}}^N$, where $p,q>1$ and $\mu >0$. We show the existence of initial data such that the corresponding solution to this system blows up in finite time $T(u_0,v_0)$ simultaneously in u and v only at one blowup point a, according to the following asymptotic dynamics:

$\{\begin{array}{cc}\hfill & u(x,t)～\mathrm{\Gamma }{[(T?t)(1+\frac{b|x?a{|}^2}{(T?t)|\log ?(T?t)|})]}^{?\frac{(p+1)}{pq?1}},\hfill \\ \hfill & v(x,t)～\gamma {[(T?t)(1+\frac{b|x?a{|}^2}{(T?t)|\log ?(T?t)|})]}^{?\frac{(q+1)}{pq?1}},\hfill \end{array}$

with $b=b(p,q,\mu )>0$ and $(\mathrm{\Gamma },\gamma )=(\mathrm{\Gamma }(p,q),\gamma (p,q))$. The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. Two major difficulties arise in the proof: the linearized operator around the profile is not self-adjoint even in the case $\mu =1$; and the fact that the case $\mu \ne 1$ breaks any symmetry in the problem. In the last section, through a geometrical interpretation of quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem, we are able to show the stability of these blowup behaviors with respect to perturbations in initial data.     

Existence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datumExistence et unicité de solutions renormalisées d'équations elliptiques non linéaires avec des termes d'ordre inférieur et données mesures

In this Note we consider a class of noncoercive nonlinear problems whose prototype is

$\text{?△}{}_{\mathrm{p}}\text{u+b(x)|?u|}{}^{\mathrm{\lambda }}\text{=μ}\text{in}\text{Ω,}\text{u=0}\text{on}\text{?Ω,}$

where $\text{Ω}$ is a bounded open subset of $\text{R}{}^{\mathrm{N}}$ (N?2), △_p is the so called p-Laplace operator (1<p<N) or a variant of it, μ is a Radon measure with bounded variation on $\text{Ω}$ or a function in $\text{L}{}^1\text{(Ω)}$, λ?0 and b belongs to the Lorentz space $\text{L}{}^{\mathrm{N,1}}\text{(Ω)}$ or to the Lebesgue space $\text{L}{}^{\mathrm{\infty }}\text{(Ω)}$. We prove existence and uniqueness of renormalized solutions. To cite this article: M.F. Betta et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 757–762.     

Une caractérisation des fonctions holomorphes injectives en analyse ultramétriqueA characterization of injective holomorphic functions in ultrametric analysis

We prove that a non constant holomorphic function f defined over an analytic subspace of $\text{C}{}_{\mathrm{p}}$ is injective if and only if

$\text{f(x)?f(y)}\text{x?y}{}^2\text{=}\text{f′(x)·f′(y)}\text{,}\text{for every distinct}\text{x}\text{and}\text{y.}$

This characterization proves the analogue, for holomorphic functions, of a conjecture of A. Escassut and M.C. Sarmant. On the other hand we give a counter-example to this conjecture, that concerns bi-analytic elements. To cite this article: J. Rivera-Letelier, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 441–446.     

Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds III: Global-in-time Strichartz estimates without lossRésolvante et mesure spectrale sur des variétés non-captives asymptotiquement hyperboliques III : Estimations de Strichartz sans perte et globales en temps

In the present paper, we investigate global-in-time Strichartz estimates without loss on non-trapping asymptotically hyperbolic manifolds. Due to the hyperbolic nature of such manifolds, the set of admissible pairs for Strichartz estimates is much larger than usual. These results generalize the works on hyperbolic space due to Anker–Pierfelice and Ionescu–Staffilani. However, our approach is to employ the spectral measure estimates, obtained in the author's joint work with Hassell, to establish the dispersive estimates for truncated/microlocalized Schrödinger propagators as well as the corresponding energy estimates. Compared with hyperbolic space, the crucial point here is to cope with the conjugate points on the manifold. Additionally, these Strichartz estimates are applied to the $L^2$ well-posedness and $L^2$ scattering for nonlinear Schrödinger equations with power-like nonlinearity and small Cauchy data.     

Problème de Monge pour n probabilitésMonge problem for n probabilities

In this Note, we generalize Gangbo–Swiech theorem for the Monge–Kantorovich problem. We study this problem for Orlicz and Köthe spaces when the function c has the form $\text{c(x}{}_1\text{,…,x}{}_{\mathrm{n}}\text{)=h(∑x}{}_{\mathrm{i}}\text{),}\text{h}$ convex on $\text{R}{}^{\mathrm{d}}\text{.}$To cite this article: H. Heinich, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 793–795.