Up to isometries,a deformation is a continuous function of its metric tensorAux isométries près,une déformation est une fonction continue de son tenseur métrique

If the Riemann–Christoffel tensor associated with a field of class $\text{C}{}^2$ of positive definite symmetric matrices of order three vanishes in a connected and simply connected open subset $\text{Ω?}\text{R}{}^3$, then this field is the metric tensor field associated with a deformation of class $\text{C}{}^3$ of the set $\text{Ω}$, uniquely determined up to isometries of $\text{R}{}^3$. We establish here that the mapping defined in this fashion is continuous, for ad hoc metrizable topologies. To cite this article: P.G. Ciarlet, F. Laurent, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 489–493.     

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).     

Sur la topologie des fibres d'une fonction définissable dans une structure o-minimale

Let $\text{V?}\text{R}{}^{\mathrm{n}}$ be a closed, non compact C² manifold and $\text{f}\text{:V→}\text{R}$ be a C² function definable in an o-minimal structure. We prove that the flow of the gradient field of f with respect to the induced riemannian metric on V embeds a non singular asymptotic critical level of f into a typical level of f. We apply this result to complex polynomials. To cite this article: D. D'Acunto, 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).     

Commuting self-adjoint partial differential operators and a group theoretic problem

In Rⁿ let Ω denote a Nikodym region (= a connected open set on which every distribution of finite Dirichlet integral is itself in $\text{L}{}^2\text{(Ω))}$. The existence of n commuting self-adjoint operators $\text{H}{}_1\text{,…, H}{}_{\mathrm{n}}\text{in}\text{L}{}^2\text{(Ω)}$ such that each H_j is a restriction of $\text{?i β}\text{βx}{}_{\mathrm{j}}$ (acting in the distribution sense) is shown to be equivalent to the existence of a set Λ ?Rⁿ such that the restrictions to Ω of the functions exp i ∑ λ_jx_j form a total orthogonal family in $\text{L}{}^2\text{(Ω)}$. If it is required, in addition, that the unitary groups generated by H₁,…, H_n act multiplicatively on $\text{L}{}^2\text{(Ω)}$, then this is shown to correspond to the requirement that Λ can be chosen as a subgroup of the additive group Rⁿ. The measurable sets Ω ?Rⁿ (of finite Lebesgue measure) for which there exists a subgroup Λ ?Rⁿ as stated are precisely those measurable sets which (after a correction by a null set) form a system of representatives for the quotient of Rⁿ by some subgroup Γ (essentially the dual of Λ).     

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).     

On rigid displacements and their relation to the infinitesimal rigid displacement lemma in three-dimensional elasticity

Let $\text{Ω}$ be an open connected subset of $\text{R}{}^3$ and let $\text{Θ}$ be an immersion from $\text{Ω}$ into $\text{R}{}^3$. It is established that the set formed by all rigid displacements of the open set $\text{Θ}\text{(Ω)}$ is a submanifold of dimension 6 and of class $\text{C}{}^{\mathrm{\infty }}$ of the space $\text{H}{}^1\text{(Ω)}$. It is also shown that the infinitesimal rigid displacements of the same set $\text{Θ}\text{(Ω)}$ span the tangent space at the origin to this submanifold. To cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

W1,p-quasiconvexity and variational problems for multiple integrals

Variational problems for the multiple integral $\text{I}{}_{\mathrm{\Omega }}\text{(u) = ∝}{}_{\mathrm{\Omega }}\text{g(▽u(x))dx}$, where $\text{Ω?R}{}^{\mathrm{m}}$ and $\text{u:Ω→R}{}^{\mathrm{n}}$ are studied. A new condition on g, called W^1,p-quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of C. B. Morrey, it being shown in particular to be necessary for sequential weak lower semicontinuity of $\text{I}{}_{\mathrm{\Omega }}$ in $\text{W}{}^{\mathrm{1,p}}\text{(Ω;R}{}^{\mathrm{n}}\text{)}$ and for the existence of minimizers for certain related integrals. Counterexamples are given concerning the weak continuity properties of Jacobians in $\text{W}{}^{\mathrm{1,p}}\text{(Ω;R}{}^{\mathrm{n}}\text{)}$, p ? n = m. An existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses.     

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).     

Une note sur les lemmes div–curl

Let $\text{Ω}$ be a strongly Lipschitz domain of $\text{R}{}^{\mathrm{n}}$ (n?2). We give endpoint versions of div–curl lemmata on $\text{Ω}$, for a given function f on $\text{Ω}$ whose gradient belongs to a Hardy space on $\text{Ω}$. To cite this article: P. Auscher et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Subextension of plurisubharmonic functions with bounded Monge–Ampère mass

Let $\text{Ω?}\text{C}{}^{\mathrm{n}}$ be a hyperconvex domain. Denote by $\text{E}{}_0\text{(Ω)}$ the class of negative plurisubharmonic functions ? on $\text{Ω}$ with boundary values 0 and finite Monge–Ampère mass on $\text{Ω.}$ Then denote by $\text{F}\text{(Ω)}$ the class of negative plurisubharmonic functions ? on $\text{Ω}$ for which there exists a decreasing sequence (?)_j of plurisubharmonic functions in $\text{E}{}_0\text{(Ω)}$ converging to ? such that $\text{sup}{}_{\mathrm{j}}\text{∫}{}_{\mathrm{\Omega }}\text{(dd}{}^{\mathrm{c}}\text{?}{}_{\mathrm{j}}\text{)}{}^{\mathrm{n}}\text{+∞.}$It is known that the complex Monge–Ampère operator is well defined on the class $\text{F}\text{(Ω)}$ and that for a function $\text{?∈}\text{F}\text{(Ω)}$ the associated positive Borel measure is of bounded mass on $\text{Ω.}$ A function from the class $\text{F}\text{(Ω)}$ is called a plurisubharmonic function with bounded Monge–Ampère mass on $\text{Ω.}$We prove that if $\text{Ω}$ and $\text{Ω}$ are hyperconvex domains with $\text{Ω?}\text{Ω}\text{?}\text{C}{}^{\mathrm{n}}$ and $\text{?∈}\text{F}\text{(Ω),}$ there exists a plurisubharmonic function $\text{?}\text{?}\text{∈}\text{F}\text{(}\text{Ω}\text{)}$ such that $\text{?}\text{?}\text{??}$ on $\text{Ω}$ and $\text{∫}{}_{\mathrm{<mtext>\Omega </mtext>}}\text{(dd}{}^{\mathrm{c}}\text{?}\text{?}\text{)}{}^{\mathrm{n}}\text{?∫}{}_{\mathrm{\Omega }}\text{(dd}{}^{\mathrm{c}}\text{?)}{}^{\mathrm{n}}\text{.}$ Such a function is called a subextension of ? to $\text{Ω}\text{.}$From this result we deduce a global uniform integrability theorem for the classes of plurisubharmonic functions with uniformly bounded Monge–Ampère masses on $\text{Ω.}$To cite this article: U. Cegrell, A. Zeriahi, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

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.     

Eigenvalue estimates for the Dirac operator and harmonic 1-forms of constant length

We prove that on a compact n-dimensional spin manifold admitting a non-trivial harmonic 1-form of constant length, every eigenvalue λ of the Dirac operator satisfies the inequality $\text{λ}{}^2\text{?}\text{n?1}\text{4(n?2)}\text{inf}{}_{\mathrm{M}}\text{Scal}$. In the limiting case the universal cover of the manifold is isometric to $\text{R}\text{×N}$ where N is a manifold admitting Killing spinors. To cite this article: A. Moroianu, L. Ornea, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

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).     

Geometric Anosov flows of dimension 5

We show that for a smooth Anosov flow on a closed five dimensional manifold, if it has C^∞ Anosov splitting and preserves a C^∞ pseudo-Riemannian metric, then up to a special time change and finite covers, it is C^∞ flow equivalent either to the suspension of a symplectic hyperbolic automorphism of $\text{T}{}^4$, or to the geodesic flow on a three dimensional hyperbolic manifold. To cite this article: Y. Fang, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Sur l'algébrisabilité locale de sous-variétés analytiques réelles génériques de Cn

We prove that the local (pseudo)group of biholomorphisms stabilizing a minimal, finitely nondegenerate real algebraic submanifold in $\text{C}{}^{\mathrm{n}}$ is a real algebraic local Lie group. We deduce necessary conditions for the local algebraizability of real analytic rigid tubes of arbitrary codimension in $\text{C}{}^{\mathrm{n}}$. To cite this article: H. Gaussier, J. Merker, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Spectral theory of commuting self-adjoint partial differential operators

Let Ω denote a connected and open subset of $\text{R}$ⁿ. The existence of n commuting self-adjoint operators H₁,…, H_n on $\text{L}{}^2\text{(Ω)}$ such that each H_j is an extension of $\text{−}\text{i∂}\text{∂x}{}_{\mathrm{j}}$ (acting on $\text{C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(Ω))}$ is shown to be equivalent to the existence of a measure μ on $\text{R}$ⁿ such that f → $\text{}\text{̂}$tf (the Fourier transform of f) is unitary from $\text{L}{}^2\text{(Ω)}$ onto Ω. It is shown that the support of μ can be chosen as a subgroup of $\text{R}$ⁿ iff H₁,…, H_n can be chosen such that the unitary groups generated by H₁,…, H_n act multiplicatively on $\text{L}{}^2\text{(Ω)}$. This happens iff Ω (after correction by a null set) forms a system of representatives for the quotient of $\text{R}$ⁿ by some subgroup, i.e., iff Ω is essentially a fundamental domain.     

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).     

Sur l'analyticité des applications CR lisses à valeurs dans un ensemble algébrique réelOn the analyticity of smooth CR maps into a real algebraic set

Let f:M→M′ be a $\text{C}{}^{\mathrm{\infty }}$-smooth CR mapping between a generic real analytic submanifold $\text{M?}\text{C}{}^{\mathrm{n}}$ and a real algebraic subset $\text{M′?}\text{C}{}^{\mathrm{n\prime }}$. We prove that if M is minimal at a point p and if M′ does not contain complex curves, then f is real-analytic at p. To cite this article: B. Coupet et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 953–956.