A nonlinear Korn inequality on a surface

Let ω be a domain in ${\mathbb{R}}^2$ and let $\mathit{\theta }:\overline{\omega }\to {\mathbb{R}}^3$ be a smooth immersion. The main purpose of this paper is to establish a “nonlinear Korn inequality on the surface $\mathit{\theta }(\overline{\omega })$”, asserting that, under ad hoc assumptions, the ${\mathit{H}}^1(\omega )$-distance between the surface $\mathit{\theta }(\overline{\omega })$ and a deformed surface is “controlled” by the ${\mathit{L}}^1(\omega )$-distance between their fundamental forms. Naturally, the ${\mathit{H}}^1(\omega )$-distance between the two surfaces is only measured up to proper isometries of ${\mathbb{R}}^3$.This inequality implies in particular the following interesting per se sequential continuity property for a sequence of surfaces. Let ${\mathit{\theta }}^k:\omega \to {\mathbb{R}}^3$, $k⩾1$, be mappings with the following properties: They belong to the space ${\mathit{H}}^1(\omega )$; the vector fields normal to the surfaces ${\mathit{\theta }}^k(\omega )$, $k⩾1$, are well defined a.e. in ω and they also belong to the space ${\mathit{H}}^1(\omega )$; the principal radii of curvature of the surfaces ${\mathit{\theta }}^k(\omega )$, $k⩾1$, stay uniformly away from zero; and finally, the fundamental forms of the surfaces ${\mathit{\theta }}^k(\omega )$ converge in ${\mathit{L}}^1(\omega )$ toward the fundamental forms of the surface $\mathit{\theta }(\overline{\omega })$ as $k\to \infty$. Then, up to proper isometries of ${\mathbb{R}}^3$, the surfaces ${\mathit{\theta }}^k(\omega )$ converge in ${\mathit{H}}^1(\omega )$ toward the surface $\mathit{\theta }(\overline{\omega })$ as $k\to \infty$.Such results have potential applications to nonlinear shell theory, the surface $\mathit{\theta }(\overline{\omega })$ being then the middle surface of the reference configuration of a nonlinearly elastic shell.     

The division problem for tempered distributions of one variable

We give a characterization, in one variable case, of those $C^{\infty }$ multipliers F such that the division problem is solvable in $S^{\prime }(\mathbb{R})$. For these functions $F\in O_M(\mathbb{R})$ we even prove that the multiplication operator $M_F(G)=FG$ has a continuous linear right inverse on $S^{\prime }(\mathbb{R})$, in contrast to what happens in the several variables case, as was shown by Langenbruch.     

Regularity of random attractors for fractional stochastic reaction–diffusion equations on Rn

We investigate the regularity of random attractors for the non-autonomous non-local fractional stochastic reaction–diffusion equations in $H^s({\mathbb{R}}^n)$ with $s\in (0,1)$. We prove the existence and uniqueness of the tempered random attractor that is compact in $H^s({\mathbb{R}}^n)$ and attracts all tempered random subsets of $L^2({\mathbb{R}}^n)$ with respect to the norm of $H^s({\mathbb{R}}^n)$. The main difficulty is to show the pullback asymptotic compactness of solutions in $H^s({\mathbb{R}}^n)$ due to the noncompactness of Sobolev embeddings on unbounded domains and the almost sure nondifferentiability of the sample paths of the Wiener process. We establish such compactness by the ideas of uniform tail-estimates and the spectral decomposition of solutions in bounded domains.     

Continuity in H1-norms of surfaces in terms of the L1-norms of their fundamental forms

The main purpose of this Note is to show how a ‘nonlinear Korn's inequality on a surface’ can be established. This inequality implies in particular the following interesting per se sequential continuity property for a sequence of surfaces. Let ω be a domain in ${\mathbb{R}}^2$, let $\mathit{\theta }:\overline{\omega }\to {\mathbb{R}}^3$ be a smooth immersion, and let ${\mathit{\theta }}^k:\overline{\omega }\to {\mathbb{R}}^3$, $k?1$, be mappings with the following properties: They belong to the space ${\mathit{H}}^1(\omega )$; the vector fields normal to the surfaces ${\mathit{\theta }}^k(\omega )$, $k?1$, are well defined a.e. in ω and they also belong to the space ${\mathit{H}}^1(\omega )$; the principal radii of curvature of the surfaces ${\mathit{\theta }}^k(\omega )$ stay uniformly away from zero; and finally, the three fundamental forms of the surfaces ${\mathit{\theta }}^k(\omega )$ converge in ${\mathit{L}}^1(\omega )$ toward the three fundamental forms of the surface $\mathit{\theta }(\omega )$ as $k\to \infty$. Then, up to proper isometries of ${\mathbb{R}}^3$, the surfaces ${\mathit{\theta }}^k(\omega )$ converge in ${\mathit{H}}^1(\omega )$ toward the surface $\mathit{\theta }(\omega )$ as $k\to \infty$. To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Critical space for the parabolic-parabolic Keller–Segel model in Rd

We study the Keller–Segel system in ${\mathbb{R}}^d$ when the chemoattractant concentration is described by a parabolic equation. We prove that the critical space, with some similarity to the elliptic case, is that the initial bacteria density satisfies $n_0\in L^a({\mathbb{R}}^d)$, $a>d/2$, and that the chemoattractant concentration satisfies $\mathrm{?}{c}_0\in L^d({\mathbb{R}}^d)$. In these spaces, we prove that small initial data give rise to global solutions that vanish as the heat equation for large times and that exhibit a regularizing effect of hypercontractivity type. To cite this article: L. Corrias, B. Perthame, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Absence de spectre absolument continu pour un opérateur d'Anderson à potentiel d'interaction générique

We present a result of absence of absolutely continuous spectrum in an interval of $\mathbb{R}$, for a matrix-valued random Schrödinger operator, acting on $L^2(\mathbb{R})?{\mathbb{R}}^N$ for an arbitrary $N?1$, and whose interaction potential is generic in the real symmetric matrices. For this purpose, we prove the existence of an interval of energies on which we have separability and positivity of the N non-negative Lyapunov exponents of the operator. The method, based upon the formalism of Fürstenberg and a result of Lie group theory due to Breuillard and Gelander, allows an explicit construction of the wanted interval of energies.     

Unique continuation for first-order systems with integrable coefficients and applications to elasticity and plasticity

Let $\Omega ?{\mathbb{R}}^N$ be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ?Ω. We show that the solution to the linear first-order system:(1)$\mathrm{?}\zeta =G\zeta ,\zeta {|}_{\Gamma }=0,$ vanishes if $G\in {\mathsf{L}}^1(\Omega ;{\mathbb{R}}^{(N\times N)\times N})$ and $\zeta \in {\mathsf{W}}^{1,1}(\Omega ;{\mathbb{R}}^N)$. In particular, square-integrable solutions ζ of (1) with $G\in {\mathsf{L}}^1\cap {\mathsf{L}}^2(\Omega ;{\mathbb{R}}^{(N\times N)\times N})$ vanish. As a consequence, we prove that:$???:{\mathsf{C}}_{°}^{\infty }(\Omega ,\Gamma ;{\mathbb{R}}^3)\to [0,\infty ),u?6\mathrm{sym}\left(\mathrm{?}u{P}^{?1}\right)6_{{\mathsf{L}}^2(\Omega )}$ is a norm if $P\in {\mathsf{L}}^{\infty }(\Omega ;{\mathbb{R}}^{3\times 3})$ with $\mathrm{Curl}P\in {\mathsf{L}}^p(\Omega ;{\mathbb{R}}^{3\times 3})$, $\mathrm{Curl}{P}^{?1}\in {\mathsf{L}}^q(\Omega ;{\mathbb{R}}^{3\times 3})$ for some $p,q>1$ with $1/p+1/q=1$ as well as $\det P?c^{+}>0$. We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let $\Phi \in {\mathsf{H}}^1(\Omega ;{\mathbb{R}}^3)$, $\Omega ?{\mathbb{R}}^3$, satisfy $\mathrm{sym}(\mathrm{?}{\Phi }^{?}\mathrm{?}\Psi )=0$ for some $\Psi \in {\mathsf{W}}^{1,\infty }(\Omega ;{\mathbb{R}}^3)\cap {\mathsf{H}}^2(\Omega ;{\mathbb{R}}^3)$ with $\det \mathrm{?}\Psi ?c^{+}>0$. Then there exists a constant translation vector $a\in {\mathbb{R}}^3$ and a constant skew-symmetric matrix $A\in \mathfrak{so}(3)$, such that $\Phi =A\Psi +a$.