Bifurcation for a class of singular elliptic problems with quadratic convection term

We study the bifurcation problem ?Δu=g(u)+λ|?u|²+μ in $\text{Ω,}\text{u=0}$ on $\text{?Ω}$, where λ,μ?0 and $\text{Ω}$ is a smooth bounded domain in $\text{R}{}^{\mathrm{N}}$. The singular character of the problem is given by the nonlinearity g which is assumed to be decreasing and unbounded around the origin. In this Note we prove that the above problem has a positive classical solution (which is unique) if and only if λ(a+μ)<λ₁, where a=lim_t→+∞g(t) and λ₁ is the first eigenvalue of the Laplace operator in $\text{H}{}^1{}_0\text{(Ω)}$. We also describe the decay rate of this solution, as well as a blow-up result around the bifurcation parameter. To cite this article: M. Ghergu, V. R?dulescu, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Levi-flat extensions from a part of the boundary

Let G be a bounded domain in $\text{C}\text{×}\text{R}$ such that $\text{G×}\text{R}\text{?}\text{C}{}^2$ is strictly pseudoconvex and U an open subset of bG. We define an open subset $\text{Ω}{}^{\mathrm{U}}$ of $\text{G}$ with the property $\text{Ω}{}^{\mathrm{U}}\text{∩bG=U}$ such that the following extension theorem holds true: for every ?∈C(U) there exist two functions $\text{Φ}{}^{\mathrm{\pm }}\text{∈C(Ω}{}^{\mathrm{U}}\text{)}$ such that Φ^±|_U=? and the graphs Γ(Φ^±) of Φ^± are Levi-flat over $\text{Ω}{}^{\mathrm{U}}\text{∩G}$. Moreover, for each $\text{Φ∈C(Ω}{}^{\mathrm{U}}\text{)}$ such that Φ|_U=? and Γ(Φ) is Levi-flat over $\text{Ω}{}^{\mathrm{U}}\text{∩G}$ one has Φ^??Φ?Φ⁺ on $\text{Ω}{}^{\mathrm{U}}$. We also show that if G is diffeomorphic to a 3-ball and U is the union of simply-connected domains each of which is contained either in the “upper” or in the “lower” part of bG (with respect to the u-direction), then $\text{Ω}{}^{\mathrm{U}}$ is the maximal domain of Levi-flat extensions for some function ?∈C(U). To cite this article: N. Shcherbina, G. Tomassini, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Régularité dans une équation de Schrödinger avec potentiel singulier à distance finie et à l'infini

In this Note we study the Schrödinger equation i?_tu+Δu+V₀u+V₁u=0 on $\text{R}{}^3\text{×(0,T)}$ with initial condition $\text{u}{}_0\text{∈{v∈H}{}^2\text{(}\text{R}{}^3\text{)}$, where V₀ is a coulombian potential, singular at finite distance and V₁ is an electric potential, possibly unbounded. Both of them may depend on space and time variables. We prove that this problem is well-posed and that the regularity of the initial data is conserved for the solution. The detailed proof will be given elsewhere (Baudouin et al., in press). To cite this article: L. Baudouin et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Détermination du potentiel dans l'équation de Schrödinger à partir de mesures sur une partie du bordDetermination of potential in Schrödinger equation from measurements on a part of the boundary

We study the Schrödinger equation iy′+Δy+qy=0 on $\text{Ω×(0,T)}$ with Dirichlet boundary data $\text{y|}{}_{\mathrm{?\Omega \times (0,T)}}$ and initial condition $\text{y|}{}_{\mathrm{\Omega \times \{0\}}}$ and we consider the inverse problem of determining the potential q(x), $\text{x∈Ω}$ when is given, where Γ₀ is an open subset of $\text{?Ω}$ satisfying an appropriate geometrical condition. The detailed proof will be given in [1]. To cite this article: L. Baudouin, J.-P. Puel, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 967–972.     

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.     

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     

Asymptotics for the blow-up boundary solution of the logistic equation with absorption

Let $\text{Ω}$ be a smooth bounded domain in $\text{R}{}^{\mathrm{N}}$. Assume that f?0 is a C¹-function on [0,∞) such that f(u)/u is increasing on (0,+∞). Let a be a real number and let b?0, b?0 be a continuous function such that b≡0 on $\text{?Ω}$. The purpose of this Note is to establish the asymptotic behaviour of the unique positive solution of the logistic problem Δu+au=b(x)f(u) in $\text{Ω}$, subject to the singular boundary condition u(x)→+∞ as $\text{dist}\text{(x,?Ω)→0}$. Our analysis is based on the Karamata regular variation theory. To cite this article: F.-C. Cîrstea, V. R?dulescu, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

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.     

Uniqueness of the blow-up boundary solution of logistic equations with absorbtion

Let $\text{Ω}$ be a smooth bounded domain in $\text{R}{}^{\mathrm{N}}$. Assume f∈C¹[0,∞) is a non-negative function such that f(u)/u is increasing on (0,∞). Let a be a real number and let b?0, $\text{b}\text{/≡0}$ be a continuous function such that b≡0 on $\text{?Ω}$. We study the logistic equation Δu+au=b(x)f(u) in $\text{Ω}$. The special feature of this work is the uniqueness of positive solutions blowing-up on $\text{?Ω}$, in a general setting that arises in probability theory. To cite this article: F.-C. C??rstea, V. R?dulescu, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 447–452.     

Doubly nonlinear parabolic equations with singular lower order term

In this paper we are concerned with positive solutions of the doubly nonlinear parabolic equation u_t=div(u^m−1|∇u|^p−2∇u)+Vu^m+p−2 in a cylinder $\text{Ω×(0,T)}$, with initial condition u(·,0)=u₀(·)⩾0 and vanishing on the parabolic boundary $\text{∂Ω×(0,T)}$. Here $\text{Ω⊂}\text{R}{}^{\mathrm{N}}$ (resp. $\text{H}{}^{\mathrm{n}}$) is a bounded domain with smooth boundary, $\text{V∈L}{}_{\mathrm{<mtext>loc</mtext>}}{}^1\text{(Ω)}$, $\text{m∈}\text{R}$, 1<p<N and m+p−2>0. The critical exponents $\text{q}{}^1$ are found and the nonexistence results are proved for $\text{q}{}^1\text{⩽m+p<3}$.     

Non existence de solutions globales de certaines équations d'ondes non linéairesNonexistence of global solutions to some nonlinear wave equations

We study the semilinear wave equation u_tt?Δu=p^?k|u|^m in $\text{R}\text{×}\text{R}{}^{\mathrm{n}}$, where p is a conformal factor approaching 0 at infinity. We prove that the solutions blow-up in finite time for small powers m, while having an arbitrarily long life-span for large m. Furthermore, we study the finite time blow-up of solutions for the class of quasilinear wave equations u_tt?Δu=p^?k|Lu|^m in $\text{R}\text{×}\text{R}{}^{\mathrm{n}}$. To cite this article: M. Aassila, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 961–966.     

Cohérence des faisceaux d'idéaux multiplicateurs avec estimations

Let $\text{Ω?}\text{C}{}^{\mathrm{n}}$ be a bounded pseudoconvex open set and let ? be a plurisubharmonic function on $\text{Ω.}$ For every positive integer m, we consider the multiplier ideal sheaf $\text{I}\text{(m?)}$ and the Hilbert space $\text{H}{}_{\mathrm{\Omega }}\text{(m?)}$ of holomorphic functions f on $\text{Ω}$ such that $\text{|f|}{}^2\text{e}{}^{\mathrm{?2m?}}$ is integrable on $\text{Ω.}$ We give an effective version, with estimates, of Nadel's result stating that the sheaf $\text{I}\text{(m?)}$ is coherent and generated by an arbitrary orthonormal basis of $\text{H}{}_{\mathrm{\Omega }}\text{(m?).}$ This result is expected to play a major part in the context of current regularizations with estimates of the Monge–Ampère masses. To cite this article: D. Popovici, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Régularité du rayon hyperbolique

Let $\text{Ω?}\text{R}{}^2$ be a bounded domain of class $\text{C}{}^{\mathrm{2+\alpha }}\text{,}\text{0<α<1}$. We show that if u is the maximal solution of Δu=4exp(2u), which tends to +∞ as $\text{(x,y)→?Ω}$, then the hyperbolic radius v=exp(?u) is of class C^2+α up to the boundary. The proof relies on new Schauder estimates for Fuchsian elliptic equations. To cite this article: S. Kichenassamy, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

A characterization of first order nonlinear partial differential operators on Sobolev spaces

Nonlinear partial differential operators $\text{G: W}{}^{\mathrm{1,p}}\text{(Ω) → L}{}^{\mathrm{q}}\text{(Ω) (1 ? p, q ∞)}$ having the form G(u) = g(u, D₁u,…, D_Nu), with g?C(R × R_N), are here shown to be precisely those operators which are local, (locally) uniformly continuous on, $\text{W}{}^{\mathrm{1,\infty }}\text{(Ω)}$, and (roughly speaking) translation invariant. It is also shown that all such partial differential operators are necessarily bounded and continuous with respect to the norm topologies of $\text{W}{}^{\mathrm{1,p}}\text{(Ω)}\text{and}\text{L}{}^{\mathrm{q}}\text{(Ω)}$.     

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).     

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).     

An estimate of the H1-norm of deformations in terms of the L1-norm of their Cauchy–Green tensors

Let $\text{Ω}$ be a bounded open connected subset of $\text{R}{}^{\mathrm{n}}$ with a Lipschitz-continuous boundary and let $\text{Θ}\text{∈}\text{C}{}^1\text{(}\text{Ω}\text{;}\text{R}{}^{\mathrm{n}}\text{)}$ be a deformation of the set $\text{Ω}$ satisfying $\text{det}\text{?Θ}\text{>0}$ in $\text{Ω}$. It is established that there exists a constant $\text{C(}\text{Θ}\text{)}$ with the following property: for each deformation $\text{Φ}\text{∈H}{}^1\text{(Ω;}\text{R}{}^{\mathrm{n}}\text{)}$ satisfying $\text{det}\text{?Φ}\text{>0}$ a.e. in $\text{Ω}$, there exist an n×n rotation matrix $\text{R}\text{=}\text{R}\text{(}\text{Φ}\text{,}\text{Θ}\text{)}$ and a vector $\text{b}\text{=}\text{b}\text{(}\text{Φ}\text{,}\text{Θ}\text{)}$ in $\text{R}{}^{\mathrm{n}}$ such that

The proof relies in particular on a fundamental ‘geometric rigidity lemma’, recently proved by G. Friesecke, R.D. James, and S. Müller. To cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On nonlinear perturbations of a periodic elliptic problem in R2 involving critical growth

We consider the equation −Δu+V(x)u=f(x,u) for $\text{x∈}\text{R}{}^2$ where $\text{V:}\text{R}{}^2\text{→}\text{R}$ is a positive potential bounded away from zero, and the nonlinearity $\text{f:}\text{R}{}^2\text{×}\text{R}\text{→}\text{R}$ behaves like exp(α|u|²) as |u|→∞. We also assume that the potential V(x) and the nonlinearity f(x,u) are asymptotically periodic at infinity. We prove the existence of at least one weak positive solution $\text{u∈H}{}^1\text{(}\text{R}{}^2\text{)}$ by combining the mountain-pass theorem with Trudinger–Moser inequality and a version of a result due to Lions for critical growth in $\text{R}{}^2$.     

Sur les équations de Lane–Emden avec opérateurs non linéaires

In this Note we consider nonnegative solutions for the nonlinear equation

$\text{M}{}^{\mathrm{+}}{}_{\mathrm{\lambda ,\Lambda }}\text{D}{}^2\text{u}\text{+|x|}{}^{\mathrm{\alpha }}\text{u}{}^{\mathrm{p}}\text{=0}$

in $\text{R}{}^{\mathrm{N}}$, where $\text{M}{}^{\mathrm{+}}{}_{\mathrm{\lambda ,\Lambda }}\text{(D}{}^2\text{u)}$ is the so called Pucci operator

$\text{M}{}^{\mathrm{+}}{}_{\mathrm{\lambda ,\Lambda }}\text{(M)=λ}\text{∑}\text{e}{}_{\mathrm{i}}\text{<0}\text{e}{}_{\mathrm{i}}\text{+Λ}\text{∑}\text{e}{}_{\mathrm{i}}\text{>0}\text{e}{}_{\mathrm{i}}\text{,}$

and the e_i are the eigenvalues of M et Λ?λ>0. We prove that if u satisfies the decreasing estimate

$\text{lim}\text{|x|→+∞}\text{|x|}{}^{\mathrm{\beta ?1}}\text{u(x)=0}$

for some β satisfying (β?1)(p?1)>2+α then u is radial. In a second time we prove that if $\text{p<}\text{N+2α+2}\text{N?2}$ and u is a nonnegative radial solution of (1), u(x)=g(r), such that g″ changes sign at most once, then u is zero. To cite this article: I. Birindelli, F. Demengel, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Multiplicativity of lp norms for matrices. II

For 1 ? p ? ∞, let

$\text{|A|}{}_{\mathrm{p}}\text{=}\text{Σ}\text{i=1}\text{m}\text{Σ}\text{j=1}\text{n}\text{, |α}{}_{\mathrm{ij}}\text{|}{}^{\mathrm{p}}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}$

, be the l_p norm of an m × n complex A = (α_ij) ?C_{m × n}. The main purpose of this paper is to find, for any p, q ? 1, the best (smallest) possible constants τ(m, k, n, p, q) and σ(m, k, n, p, q) for which inequalities of the form

$\text{|AB|}{}_{\mathrm{p}}\text{? τ(m, k, n, p, q) |A|}{}_{\mathrm{p}}\text{|B|}{}_{\mathrm{q}}\text{, |AB|}{}_{\mathrm{p}}\text{? σ (m, k, n, p, q)|A|}{}_{\mathrm{q}}\text{|B|}{}_{\mathrm{p}}$

hold for all A?C_{m × k}, B?C_{k × n}. This leads to upper bounds for inner products on C^k and for ordinary l_p operator norms on C_{m × n}.