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).     

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.     

The question of interior blow-up points for an elliptic Neumann problem: the critical case

In contrast with the subcritical case, we prove that for any bounded domain $\text{Ω}$ in $\text{R}{}^3$, the Neumann elliptic problem with critical nonlinearity $\text{−Δu+μu=u}{}^5\text{,}\text{u>0}\text{in}\text{Ω;}\text{∂u/∂ν=0}\text{on}\text{∂Ω}$ has no solution blowing up at only interior points as μ goes to infinity.     

Higher order energy expansions for some singularly perturbed Neumann problems

We consider the following singularly perturbed semilinear elliptic problem:

$\text{ε}{}^2\text{Δ}\text{u?u+u}{}^{\mathrm{p}}\text{=0}\text{in}\text{Ω,}\text{u>0}\text{in}\text{Ω}\text{and}\text{?u}\text{?ν}\text{=0}\text{on}\text{?Ω,}$

where $\text{Ω}$ is a bounded smooth domain in $\text{R}{}^{\mathrm{N}}$, ε>0 is a small constant and p is a subcritical exponent. Let $\text{J}{}_{\mathrm{\epsilon }}\text{[u]:=∫}{}_{\mathrm{\Omega }}\text{(}\text{ε}{}^2\text{2}\text{|?u|}{}^2\text{+}\text{1}\text{2}\text{u}{}^2\text{?}\text{1}\text{p+1}\text{u}{}^{\mathrm{p+1}}\text{)}\text{d}\text{x}$ be its energy functional, where $\text{u∈H}{}^1\text{(Ω)}$. Ni and Takagi proved that for a single boundary spike solution u_ε, the following asymptotic expansion holds

$\text{J}{}_{\mathrm{\epsilon }}\text{[u}{}_{\mathrm{\epsilon }}\text{]=ε}{}^{\mathrm{N}}\text{1}\text{2}\text{I[w]?c}{}_1\text{εH(P}{}_{\mathrm{\epsilon }}\text{)+}\text{o}\text{(ε)}\text{,}$

where c₁>0 is a generic constant, P_ε is the unique local maximum point of u_ε and H(P_ε) is the boundary mean curvature function. In this Note, we obtain the following higher order expansion of J_ε[u_ε]:

$\text{J}{}_{\mathrm{\epsilon }}\text{[u}{}_{\mathrm{\epsilon }}\text{]=ε}{}^{\mathrm{N}}\text{1}\text{2}\text{I[w]?c}{}_1\text{εH(P}{}_{\mathrm{\epsilon }}\text{)+ε}{}^2\text{[c}{}_2\text{(H(P}{}_{\mathrm{\epsilon }}\text{))}{}^2\text{+c}{}_3\text{R(P}{}_{\mathrm{\epsilon }}\text{)]+}\text{o}\text{(ε}{}^2\text{)}\text{,}$

where c₂, c₃ are generic constants and R(P_ε) is the Ricci scalar curvature at P_ε. In particular c₃>0. Applications of this expansion will be given. To cite this article: J. Wei, M. Winter, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

On a nonlinear fourth-order elliptic equation involving the critical Sobolev exponent

In this paper we use an algebraic topological argument due to Bahri and Coron to show how the topology of the domain influences the existence of positive solutions of the following problem involving the bilaplacian operator with the critical Sobolev exponent$\text{Δ}{}^2\text{u=u}{}^{\mathrm{n+4/(n-4)}}\text{in}\text{Ω,}\text{u>0}\text{in}\text{Ω,}\text{u=}\text{Δ}\text{u=0}\text{on}\text{∂Ω,}$where $\text{Ω}$ is a bounded domain of $\text{R}{}^{\mathrm{n}}\text{(n⩾5)}$ with a smooth boundary $\text{∂Ω}$.     

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.     

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.     

Über die nichtlineare diophantische Approximation von quadratischen,algebraischen Zahlen

In this paper I prove the following Theorem: Let α be a quadratic irrational number and let ? > 0, $\text{t =}\text{1}\text{1728}$. Then there exists an effectively computable, positive constant c depending on α and ? such that

$\text{6q}{}^2\text{α6 < c(}\text{log}\text{q)}{}^{\mathrm{t??}}\text{q}{}^{\mathrm{?2}}$

has no solutions in natural numbers q.     

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).     

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.     

Diffusion with interactions between two types of particles and Pressureless gas equations

We construct two d-dimensional independent diffusions $\text{X}{}_{\mathrm{t}}{}^{\mathrm{a}}\text{=a+∫}{}_0{}^{\mathrm{t}}\text{u(X}{}_{\mathrm{s}}{}^{\mathrm{a}}\text{,s)}\text{d}\text{s+νB}{}_{\mathrm{t}}{}^{\mathrm{a}}\text{,}\text{X}{}_{\mathrm{t}}{}^{\mathrm{b}}\text{=b+∫}{}_0{}^{\mathrm{t}}\text{u(X}{}_{\mathrm{s}}{}^{\mathrm{b}}\text{,s)}\text{d}\text{s+νB}{}_{\mathrm{t}}{}^{\mathrm{b}}$, with the same viscosity ν≠0 and the same drift u(x,t)=(pρ_t^a(x)v₁+(1?p)ρ_t^b(x)v₂)/(pρ_t^a(x)+(1?p)ρ_t^b(x)), where ρ_t^a,ρ_t^b are respectively the density of X_t^a and X_t^b. Here $\text{a,b,v}{}_1\text{,v}{}_2\text{∈}\text{R}{}^{\mathrm{d}}$ and p∈(0,1) are given. We show that $\text{(ρ}{}_{\mathrm{t}}\text{(x)=pρ}{}_{\mathrm{t}}{}^{\mathrm{a}}\text{(x)+(1?p)ρ}{}_{\mathrm{t}}{}^{\mathrm{b}}\text{(x),u(x,t):}\text{t?0,}\text{x∈}\text{R}{}^{\mathrm{d}}\text{)}$ is the unique weak solution of the following pressureless gas system

such that $\text{ρ}{}_{\mathrm{t}}\text{(x)}\text{d}\text{x→pδ}{}_{\mathrm{a}}\text{+(1?p)δ}{}_{\mathrm{b}}\text{,}\text{u(x,t)ρ}{}_{\mathrm{t}}\text{(x)}\text{d}\text{x→pv}{}_1\text{δ}{}_{\mathrm{a}}\text{+(1?p)v}{}_2\text{δ}{}_{\mathrm{b}}$ as t→0+. To cite this article: A. Dermoune, S. Filali, C. R. Acad. Sci. Paris, Ser. I 337 (2003).