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.     

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).     

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).     

On the normal form of a system of differential equations with nilpotent linear part

We consider prenormal forms associated to generic perturbations of the system $\text{x}\text{?}\text{=2y,}\text{y}\text{?}\text{=3x}{}^2$. It is known that they have a formal normal form $\text{x}\text{?}\text{=2y+2x}\text{Δ}{}^1\text{,}\text{y}\text{?}\text{=3x}{}^2\text{+3y}\text{Δ}{}^1$, where $\text{Δ}{}^1\text{=x+A}{}_0\text{(y}{}^2\text{?x}{}^3\text{)}$ [Differential Equations 158 (1) (1999) 152–173]. We show that the series A₀ and the normalizing transformations are divergent, but 1-summable. To cite this article: M. Canalis-Durand, R. Schäfke, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Processus de Bickel–Rosenblatt pondéré et tests d'ajustement

The goal of this work is to establish the limit distribution of the process

$\text{I}{}_{\mathrm{n}}\text{(W):=}\text{∫}\text{A}\text{f}{}_{\mathrm{n}}\text{(x)?Ef}{}_{\mathrm{n}}\text{(x)}{}^2\text{W(x)}\text{d}\text{x,}\text{W∈}\text{W}\text{,}$

where $\text{W}$ is a class of weight functions W, f_n is the kernel density estimator of the density f and A is a Borelian subset of $\text{R}$. We apply this result to derive new statistics to test goodness-of-fit of the density function f. Under some local alternatives, these new tests are more powerful than the usual Bickel–Rosenblatt one. To cite this article: F. Chebana, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Existence of solutions for nonlinear elliptic degenerate equations

In this paper, we study the problem$\text{−div}\text{a(x,u,}\text{∇}\text{u)−div}\text{φ(u)+g(x,u)=f}\text{in}\text{Ω}$in the setting of the weighted sobolev space $\text{W}{}_0{}^{\mathrm{1,p}}\text{(Ω,ν)}$. The main novelty of our work is L^∞ estimates on the solutions, and the existence of a weak and renormalized solution.