Expression of Dirichlet boundary conditions in terms of the strain tensor in linearized elasticity

In a previous work, it was shown how the linearized strain tensor field $\mathbf{e}:=\frac{1}{2}(\mathbf{?}{\mathit{u}}^T+\mathbf{?}\mathit{u})\in {\mathbb{L}}^2(\Omega )$ can be considered as the sole unknown in the Neumann problem of linearized elasticity posed over a domain $\Omega ?{\mathbb{R}}^3$, instead of the displacement vector field $\mathit{u}\in {\mathit{H}}^1(\Omega )$ in the usual approach. The purpose of this Note is to show that the same approach applies as well to the Dirichlet–Neumann problem. To this end, we show how the boundary condition $\mathit{u}=\mathbf{0}$ on a portion ${\Gamma }_0$ of the boundary of Ω can be recast, again as boundary conditions on ${\Gamma }_0$, but this time expressed only in terms of the new unknown $\mathbf{e}\in {\mathbb{L}}^2(\Omega )$.     

Recovery of sparsest signals via ℓq-minimization

In this paper, it is proved that every s-sparse vector $\mathbf{x}\in {\mathbb{R}}^n$ can be exactly recovered from the measurement vector $\mathbf{z}=\mathbf{Ax}\in {\mathbb{R}}^m$ via some ${?}^q$-minimization with $0<q?1$, as soon as each s-sparse vector $\mathbf{x}\in {\mathbb{R}}^n$ is uniquely determined by the measurement z. Moreover it is shown that the exponent q in the ${?}^q$-minimization can be so chosen to be about $0.6796\times (1?{\delta }_{2s}(\mathbf{A}))$, where ${\delta }_{2s}(\mathbf{A})$ is the restricted isometry constant of order 2s for the measurement matrix A.     

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.     

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).     

Approximation of maximum of Gaussian random fields

This contribution is concerned with Gumbel limiting results for supremum ${\mathbb{M}}_{\mathit{n}}={\sup }_{\mathit{t}\in [\mathbf{0},{\mathit{T}}_{\mathit{n}}]}?|X_{\mathit{n}}(\mathit{t})|$ with $X_{\mathit{n}},\mathit{n}\in {\mathbb{N}}^2$ centered Gaussian random fields with continuous trajectories. We show first the convergence of a related point process to a Poisson point process thereby extending previous results obtained in [8] for Gaussian processes. Furthermore, we derive Gumbel limit results for ${\mathbb{M}}_{\mathit{n}}$ as $\mathit{n}\to \infty$ and show a second-order approximation for $\mathbb{E}{\{{\mathbb{M}}_{\mathit{n}}^p\}}^{1/p}$ for any $p\ge 1$.     

A continuum of periodic solutions to the planar four-body problem with two pairs of equal masses

In this paper, we apply the variational method with Structural Prescribed Boundary Conditions (SPBC) to prove the existence of periodic and quasi-periodic solutions for the planar four-body problem with two pairs of equal masses $m_1=m_3$ and $m_2=m_4$. A path $q(t)$ on $[0,T]$ satisfies the SPBC if the boundaries $q(0)\in \mathbf{A}$ and $q(T)\in \mathbf{B}$, where A and B are two structural configuration spaces in ${({\mathbf{R}}^2)}^4$ and they depend on a rotation angle $\theta \in (0,2\pi )$ and the mass ratio $\mu =\frac{m_2}{m_1}\in {\mathbf{R}}^{+}$.We show that there is a region $\mathrm{\Omega }?(0,2\pi )\times R^{+}$ such that there exists at least one local minimizer of the Lagrangian action functional on the path space satisfying the SPBC $\{q(t)\in H^1([0,T],{({\mathbf{R}}^2)}^4)|q(0)\in \mathbf{A},q(T)\in \mathbf{B}\}$ for any $(\theta ,\mu )\in \mathrm{\Omega }$. The corresponding minimizing path of the minimizer can be extended to a non-homographic periodic solution if θ is commensurable with π or a quasi-periodic solution if θ is not commensurable with π. In the variational method with the SPBC, we only impose constraints on the boundary and we do not impose any symmetry constraint on solutions. Instead, we prove that our solutions that are extended from the initial minimizing paths possess certain symmetries.The periodic solutions can be further classified as simple choreographic solutions, double choreographic solutions and non-choreographic solutions. Among the many stable simple choreographic orbits, the most extraordinary one is the stable star pentagon choreographic solution when $(\theta ,\mu )=(\frac{4\pi }{5},1)$. Remarkably the unequal-mass variants of the stable star pentagon are just as stable as the equal mass choreographies.     

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 $\Phi =A\Psi +a$.     

On some inequalities of Bourgain,Brezis, Maz'ya,and Shaposhnikova related to L1 vector fields

Bourgain and Brezis established, for maps $f\in L^n({\mathbb{T}}^n)$ with zero average, the existence of a solution $\mathit{Y}\in W^{1,n}\cap L^{\infty }$ of (1) $\mathrm{div}\mathit{Y}=f$. Maz'ya proved that if, in addition, $f\in H^{n/2?1}({\mathbb{T}}^n)$, then (1) can be solved in $H^{n/2}\cap L^{\infty }$. Their arguments are quite different. We present an elementary property of fundamental solutions of the biharmonic operator in two dimensions. This property unifies, in two dimensions, the two approaches, and implies another (apparently unrelated) estimate of Maz'ya and Shaposhnikova. We discuss higher dimensional analogs of the above results.     

On the sub poly-harmonic property for solutions to (−Δ)pu < 0 in Rn

In this note, we mainly study the relation between the sign of ${(?\mathrm{\Delta })}^{p}u$ and ${(?\mathrm{\Delta })}^{p?i}u$ in ${\mathbb{R}}^n$ with $p?2$ and $n?2$ for $1?i?p?1$. Given the differential inequality ${(?\mathrm{\Delta })}^{p}u<0$, first we provide several sufficient conditions so that ${(?\mathrm{\Delta })}^{p?1}u<0$ holds. Then we provide conditions such that ${(?\mathrm{\Delta })}^{i}u<0$ for all $i=1,2,\dots ,p?1$, which is known as the sub poly-harmonic property for u. In the last part of the note, we revisit the super poly-harmonic property for solutions to ${(?\mathrm{\Delta })}^{p}u={\mathrm{e}}^{2pu}$ and ${(?\mathrm{\Delta })}^{p}u=u^q$ with $q>0$ in ${\mathbb{R}}^n$.     

Higher genus Kashiwara–Vergne problems and the Goldman–Turaev Lie bialgebra

We define a family ${\mathrm{KV}}^{(g,n+1)}$ of Kashiwara–Vergne problems associated with compact connected oriented 2-manifolds of genus g with $n+1$ boundary components. The problem ${\mathrm{KV}}^{(0,3)}$ is the classical Kashiwara–Vergne problem from Lie theory. We show the existence of solutions to ${\mathrm{KV}}^{(g,n+1)}$ for arbitrary g and n. The key point is the solution to ${\mathrm{KV}}^{(1,1)}$ based on the results by B. Enriquez on elliptic associators. Our construction is motivated by applications to the formality problem for the Goldman–Turaev Lie bialgebra ${\mathfrak{g}}^{(g,n+1)}$. In more detail, we show that every solution to ${\mathrm{KV}}^{(g,n+1)}$ induces a Lie bialgebra isomorphism between ${\mathfrak{g}}^{(g,n+1)}$ and its associated graded $\mathrm{gr}{\mathfrak{g}}^{(g,n+1)}$. For $g=0$, a similar result was obtained by G. Massuyeau using the Kontsevich integral. For $g\ge 1$, $n=0$, our results imply that the obstruction to surjectivity of the Johnson homomorphism provided by the Turaev cobracket is equivalent to the Enomoto–Satoh obstruction.     

On the irreducible action of PSL(2,R) on the 3-dimensional Einstein universe

In this paper, we study the irreducible representation of $\text{PSL}(2,\mathbb{R})$ in $\text{PSL}(5,\mathbb{R})$. This action preserves a quadratic form with signature $(2,3)$. Thus, it acts conformally on the 3-dimensional Einstein universe $\mathbb{E}{\mathrm{in}}^{1,2}$. We describe the orbits induced in $\mathbb{E}{\mathrm{in}}^{1,2}$ and its complement in $\mathbb{R}{\mathbb{P}}^4$. This work completes the study in [2], and is one element of the classification of cohomogeneity one actions on $\mathbb{E}{\mathrm{in}}^{1,2}$[5].