Publications

2013

• The classification of four-end solutions to the Allen-Cahn equation on the plane
  
  • Kowalczyk Michal
  • Liu Yong
  • Pacard Frank
  Analysis & PDE, Mathematical Sciences Publishers, 2013, 6 (7), pp.1675-1718. (10.2140/apde.2013.6.1675)
  DOI : 10.2140/apde.2013.6.1675
• Compact homogeneous lcK manifolds are Vaisman
  
  • Gauduchon Paul
  • Moroianu Andrei
  • Ornea Liviu
  , 2013.
• Utilisation d'OpenFlow et des modules Splite Data Plane de DELL pour traiter le DUID-MAC-spoofing des requêtes DHCPv6
  
  • Bruyère Marc
  • Delavennat David
  , 2013. IPv6 a longtemps été associé à un mécanisme d'auto-configuration sans-état (Router Advertisements). La mise à disposition tardive d'une implémentation de protocole d'attribution d'adresses IPv6, avec-état, par l'ISC, a été un frein à son déploiement sur un parc de postes clients " administrés ". Cependant, l'arrivée du protocole DHCPv6 ne se fait pas sans soulever de nouvelles difficultés opérationnelles. Dans les environnements à double pile IP, le fait que DHCPv4 et DHCPv6 n'utilisent pas les mêmes attributs pour identifier les objets IP (MAC dans le premier cas, DUID dans le second), complique la tâche des ASR (plusieurs variantes de DUID coexistent) et rend problématique la corrélation des adressages au sein des journaux d'activités réseau. Nous souhaitons pouvoir utiliser notre infrastructure DHCPv6 ISC préexistante, qui, ne supporte que les attributs DUID, tout en référençant nos postes clients par leurs adresses MAC, peu importe la version du protocole DHCP. Cette contrainte peut être résolue grâce au module Split Data Plane embarquant un Network Processor Octeon dans les commutateurs PowerConnect DELL. Ces modules sont pilotables via OpenFlow et permettent de supporter un large nombre de fonctionnalités réseau. Le Software Defined Network déporte le plan de contrôle du réseau sur un serveur externe ce qui permet de lever une part importante des contraintes des architectures actuelles basées sur les protocoles de contrôle. Le module Split Data Plane ouvre aussi le concept de fonctionnalité hardware modulaire et open source.
• Inverse coefficient problem for Grushin-type parabolic operators
  
  • Beauchard Karine
  • Cannarsa Piermarco
  , 2013. The approach to Lipschitz stability for uniformly parabolic equations introduced by Imanuvilov and Yamamoto in 1998 based on Carleman estimates, seems hard to apply to the case of Grushin-type operators studied in this paper. Indeed, such estimates are still missing for parabolic operators degenerating in the interior of the space domain. Nevertheless, we are able to prove Lipschitz stability results for inverse coefficient problems for such operators, with locally distributed measurements in arbitrary space dimension. For this purpose, we follow a strategy that combines Fourier decomposition and Carleman inequalities for certain heat equations with nonsmooth coefficients (solved by the Fourier modes).
• Contrôle d'équations de Schrödinger et d'équations paraboliques dégénérées singulières
  
  • Morancey Morgan
  , 2013. Ce mémoire présente les travaux réalisés au cours de ma thèse sur le contrôle d'équations aux dérivées partielles. La première partie est consacrée à l'étude d'équations de Schrödinger bilinéaires unidimensionnelles autour de deux axes : la non contrôlabilité en temps petit avec des contrôles petits et la contrôlabilité simultanée. On établit un cadre pour lequel, bien que la vitesse de propagation du système soit infinie, la contrôlabilité exacte locale avec des contrôles petits est vérifiée si et seulement si le temps est suffisamment grand. Ces résultats, basés sur la coercivité d'une forme quadratique associée au développement de l'état à l'ordre deux, sont étendus dans le contexte de la contrôlabilité simultanée. On montre alors, en utilisant la méthode du retour de J.-M.~Coron, des résultats de contrôle exact local simultané pour deux ou trois équations, à phase globale et/ou à retard global près. La trajectoire de référence utilisée est construite via des résultats de contrôle partiel. En utilisant un argument de perturbation, on étend cette idée pour montrer la contrôlabilité exacte globale d'un nombre quelconque d'équations sans hypothèse sur le potentiel. Dans la deuxième partie, on prend en compte dans le modèle un terme supplémentaire quadratique en le contrôle. Ce terme, dit de polarisabilité, généralement négligé, présente un intérêt physique dans la modélisation, mais aussi mathématique dans le cas où le terme bilinéaire est insuffisant pour conclure à la contrôlabilité. En dimension quelconque, on construit des contrôles explicites réalisant la contrôlabilité approchée de l'état fondamental. En adaptant conjointement l'argument de perturbation précédent et certains résultats du contrôle bilinéaire, on prouve la contrôlabilité globale exacte de l'équation de Schrödinger avec polarisabilité 1D. La dernière partie de ce mémoire est consacrée à l'étude de la continuation unique pour un opérateur de type Grushin sur un rectangle 2D. Cet opérateur présente une singularité et une dégénérescence sur un segment séparant le domaine en deux composantes. On donne une condition nécessaire et suffisante sur le coefficient du potentiel singulier pour obtenir la continuation unique.
• Stability issues in the quasineutral limit of the one-dimensional Vlasov-Poisson equation
  
  • Han-Kwan Daniel
  • Hauray Maxime
  , 2015, pp.1101-1152. This work is concerned with the quasineutral limit of the one-dimensional Vlasov-Poisson equation, for initial data close to stationary homogeneous profiles. Our objective is threefold: first, we provide a proof of the fact that the formal limit does not hold for homogeneous profiles that satisfy the Penrose instability criterion. Second, we prove on the other hand that the limit is true for homogeneous profiles that satisfy some monotonicity condition, together with a symmetry condition. We handle the case of well-prepared as well as ill- prepared data. Last, we study a stationary boundary-value problem for the formal limit, the so-called quasineutral Vlasov equation. We show the existence of numerous stationary states, with a lot of freedom in the construction (compared to that of BGK waves for Vlasov-Poisson): this illustrates the degeneracy of the limit equation. (10.1007/s00220-014-2217-4)
  DOI : 10.1007/s00220-014-2217-4
• Preface [Special issue in memory of Yahya Ould Hamidoune]
  
  • Plagne Alain
  • Serra Oriol
  • Zemor Gilles
  European Journal of Combinatorics, Elsevier, 2013, 34 (8), pp.1205-1206. (10.1016/j.ejc.2013.05.004)
  DOI : 10.1016/j.ejc.2013.05.004
• Yahya Ould Hamidoune’s mathematical journey: A critical review of his work
  
  • Plagne Alain
  • Serra Oriol
  • Zemor Gilles
  European Journal of Combinatorics, Elsevier, 2013, 34 (8), pp.1207-1222. We present the mathematical work of Yahya Ould Hamidoune, emphasizing his main achievements, notably in graph theory and additive combinatorics. (10.1016/j.ejc.2013.05.005)
  DOI : 10.1016/j.ejc.2013.05.005
• Large restricted sumsets in general abelian group
  
  • Hamidoune Yahya Ould
  • Lopez Susana C.
  • Plagne Alain
  European Journal of Combinatorics, Elsevier, 2013, 34 (8), pp.1348-1364. Let A, B and S be three subsets of a finite Abelian group G. The restricted sumset of A and B with respect to S is defined as A\wedge^{S} B= {a+b: a in A, b in B and a-b not in S}. Let L_S=max_{z in G}| {(x,y): x,y in G, x+y=z and x-y in S}|. A simple application of the pigeonhole principle shows that |A|+|B|>|G|+L_S implies A\wedge^S B=G. We then prove that if |A|+|B|=|G|+L_S then |A\wedge^S B|>= |G|-2|S|. We also characterize the triples of sets (A,B,S) such that |A|+|B|=|G|+L_S and |A\wedge^S B|= |G|-2|S|. Moreover, in this case, we also provide the structure of the set G\setminus (A\wedge^S B). (10.1016/j.ejc.2013.05.020)
  DOI : 10.1016/j.ejc.2013.05.020
• Degenerate parabolic operators of Kolmogorov type with a geometric control condition.
  
  • Beauchard Karine
  • Helffer Bernard
  • Henry Raphael
  • Robbiano Luc
  , 2013, pp.(submitted). We consider Kolmogorov-type equations on a rectangle domain, that combine diff usion in variable v and transport in variable x at speed v^m (m is an integer), with Dirichlet boundary conditions in v. We study the null controllability of this equation with a distributed control as source term, localized on a subset of the rectangle domain. In dimension one, when the control acts on a horizontal strip, that does no contain {x=0}, then the system is null controllable in any time T > 0 when m= 1, and only in large time T > T_min > 0 when m= 2 (see [10]). In this article, we prove that, when m > 2, the system is not null controllable (whatever T is) in this confi guration. This is due to the diff usion weakening produced by the first order term. When the control acts on a vertical strip, we investigate the null controllability on a toy model, where (d/dx, x in the 1D torus) is replaced by ( \sqrt{-\Delta},x in \Omega_1), and \Omega_1 is an open subset of R^N. As the original system, this toy model satis fies the controllability properties listed above. We prove that, for m=1 and for appropriate domains \Omega_1, then null controllability does not hold (whatever T > 0 is), when the control acts on a vertical strip. Thus, a geometric control condition is required for the null controllability of this toy model. This indicates that a geometric control condition may be necessary for the original model too.
• Serrin's overdetermined problem and constant mean curvature surfaces
  
  • del Pino Manuel
  • Pacard Frank
  • Wei Juncheng
  , 2013.
• Autour de l'irrégularité des connexions méromorphes.
  
  • Teyssier Jean-Baptiste
  , 2013. Les deux premières parties de cette thèse s'inscrivent dans le contexte des analogies entre l'irrégularité pour les connexions méromorphes et la ramification sauvage des faisceaux l-adiques. On y développe l'analogue pour les connexions méromorphes de la construction d'Abbes et Saito, tout d'abord dans le cas d'un trait, puis en dimension supérieure. En première partie, on prouve une formule explicite reliant les invariants produits par la construction d'Abbes et Saito appliquée à un module différentiel M aux parties les plus polaires des formes différentielles intervenant dans la décomposition de Levelt-Turrittin de M. Dans la seconde, on généralise en dimension supérieure l'observation issue de la première partie que sur un corps algébriquement clos, les modules produits par la construction d'Abbes et Saito sont des sommes finies de modules exponentiels associés à des formes linéaires. Dans la dernière partie de cette thèse, on montre que le lieu des points stables d'une connexion méromorphe M le long d'un diviseur lisse est un sous-ensemble de l'intersection des lieux où les faisceaux d'irrégularité de M et End M sont des systèmes locaux. Enfin, on discute d'une stratégie d'attaque de l'inclusion réciproque, et on démontre à l'aide d'un critère d'André pour les points stables que si elle est vraie en dimension 2, alors elle est vraie en toute dimension.
• Variety of power sums and divisors in the moduli space of cubic fourfolds
  
  • Ranestad Kristian
  • Voisin Claire
  , 2013. We show that a cubic fourfold F that is apolar to a Veronese surface has the property that its variety of power sums VSP(F,10) is singular along a K3 surface of genus 20. We prove that these cubics form a divisor in the moduli space of cubic fourfolds and that this divisor is not a Noether-Lefschetz divisor. We use this result to prove that there is no nontrivial Hodge correspondence between a very general cubic and its VSP.
• A uniform open image theorem for ℓ-adic representations, II
  
  • Cadoret Anna
  • Tamagawa Akio
  Duke Mathematical Journal, Duke University Press, 2013, 162 (12). Let k be a field finitely generated over Q and let X be a curve over k. Fix a prime . A representation ρ : π 1 (X) → GLm(Z ) is said to be geometrically Lie perfect if any open subgroup of ρ(π 1 (X k )) has finite abelianization. Let G denote the image of ρ. Any closed point x on X induces a splitting x : Γ κ(x) := π 1 (Spec(κ(x))) → π 1 (X κ(x) ) of the restriction epimorphism π 1 (X κ(x) ) → Γ κ(x) (here, κ(x) denotes the residue field of X at x) so one can define the closed subgroup Gx :<p>The main result of this paper is the following uniform open image theorem. Under the above assumptions, for any geometrically Lie perfect representation ρ : π 1 (X) → GLm(Z ) and any integer d ≥ 1, the set X ρ,d of all closed points x ∈ X such that Gx is not open in G and [κ(x) : k] ≤ d is finite and there exists an integer</p><p>A key ingredient of our proof is that, for any integer γ ≥ 1 there exist an integer ν = ν(γ) ≥ 1 such that, given any projective system</p><p>curves (over an algebraically closed field of characteristic 0) with the same gonality γ and with Y n+1 → Yn a Galois cover of degree &gt; 1, one can construct a projective system of genus 0 curves</p><p>This, together with the case for d = 1 (which is the main result of part I of this paper), gives the proof for general d.</p><p>Our method also yields the following unconditional variant of our main result. With the above assumptions on k and X, for any -adic representation ρ : π 1 (X) → GLm(Z ) and integer d ≥ 1, the set of all closed points x ∈ X such that Gx is of codimension at least 3 in G and [κ(x) : k] ≤ d is finite.</p> (10.1215/00127094-2323013)
  DOI : 10.1215/00127094-2323013
• Inertial-sensor bias estimation from brightness/depth images and based on SO(3)-invariant integro/partial-differential equations on the unit sphere
  
  • Zarrouati-Vissiere Nadege
  • Beauchard Karine
  • Rouchon Pierre
  , 2013. Constant biases associated to measured linear and angular velocities of a moving object can be estimated from measurements of a static scene by embedded brightness and depth sensors. We propose here a Lyapunov-based observer taking advantage of the SO(3)-invariance of the partial differential equations satisfied by the measured brightness and depth fields. The resulting asymptotic observer is governed by a non-linear integro/partial differential system where the two independent scalar variables indexing the pixels live on the unit sphere of the 3D Euclidian space. The observer design and analysis are strongly simplified by coordinate-free differential calculus on the unit sphere equipped with its natural Riemannian structure. The observer convergence is investigated under C^1 regularity assumptions on the object motion and its scene. It relies on Ascoli-Arzela theorem and pre-compactness of the observer trajectories. It is proved that the estimated biases converge towards the true ones, if and only if, the scene admits no cylindrical symmetry. The observer design can be adapted to realistic sensors where brightness and depth data are only available on a subset of the unit sphere. Preliminary simulations with synthetic brightness and depth images (corrupted by noise around 10%) indicate that such Lyapunov-based observers should be robust and convergent for much weaker regularity assumptions.
• Inverse source problem and null controllability for multidimensional parabolic operators of Grushin type
  
  • Beauchard Karine
  • Cannarsa Piermarco
  • Yamamoto Masahiro
  Inverse Problems, IOP Publishing, 2014, 30 (2), pp.025006. The approach to Lipschitz stability for uniformly parabolic equations introduced by Imanuvilov and Yamamoto in 1998, based on Carleman estimates, seems hard to apply to the case of Grushin-type operators of interest to this paper. Indeed, such estimates are still missing for parabolic operators degenerating in the interior of the space domain. Nevertheless, we are able to prove Lipschitz stability results for inverse source problems for such operators, with locally distributed measurements in arbitrary space dimension. For this purpose, we follow a mixed strategy which combines the appraoch due to Lebeau and Robbiano, relying on Fourier decomposition, with Carleman inequalities for heat equations with nonsmooth coefficients (solved by the Fourier modes). As a corollary, we obtain a direct proof of the observability of multidimensional Grushin-type parabolic equations, with locally distributed observations, which is equivalent to null controllability with locally distributed controls. (10.1088/0266-5611/30/2/025006)
  DOI : 10.1088/0266-5611/30/2/025006
• Explicit approximate controllability of the Schrödinger equation with a polarizability term
  
  • Morancey Morgan
  Mathematics of Control, Signals, and Systems, Springer Verlag, 2013, 25 (3), pp.407-432. We consider a controlled Schrödinger equation with a dipolar and a polarizability term, used when the dipolar approximation is not valid. The control is the amplitude of the external electric field, it acts non linearly on the state. We extend in this infinite dimensional framework previous techniques used by Coron, Grigoriu, Lefter and Turinici for stabilization in finite dimension. We consider a highly oscillating control and prove the semi-global weak $H^2$ stabilization of the averaged system using a Lyapunov function introduced by Nersesyan. Then it is proved that the solutions of the Schrödinger equation and of the averaged equation stay close on every finite time horizon provided that the control is oscillating enough. Combining these two results, we get approximate controllability to the ground state for the polarizability system. (10.1007/s00498-012-0102-2)
  DOI : 10.1007/s00498-012-0102-2
• Solutions of semilinear elliptic equations in tubes
  
  • Pacard Frank
  • Pacella Filomena
  • Sciunzi Berardino
  The Journal of Geometric Analysis, Springer, 2014, 24 (1), pp.445-471. Given a smooth compact k-dimensional manifold \Lambda embedded in $\mathbb {R}^m$, with m\geq 2 and 1\leq k\leq m-1, and given \epsilon>0, we define B_\epsilon (\Lambda) to be the geodesic tubular neighborhood of radius \epsilon about \Lambda. In this paper, we construct positive solutions of the semilinear elliptic equation \Delta u + u^p = 0 in B_\epsilon (\Lambda) with u = 0 on \partial B_\epsilon (\Lambda), when the parameter \epsilon is chosen small enough. In this equation, the exponent p satisfies either p > 1 when n:=m-k \leq 2 or p\in (1, \frac{n+2}{n-2}) when n>2. In particular p can be critical or supercritical in dimension m\geq 3. As \epsilon tends to zero, the solutions we construct have Morse index tending to infinity. Moreover, using a Pohozaev type argument, we prove that our result is sharp in the sense that there are no positive solutions for p>\frac{n+2}{n-2}, n\geq 3, if \epsilon is sufficiently small. (10.1007/s12220-012-9342-0)
  DOI : 10.1007/s12220-012-9342-0
• Charles Favre - Application to complex dynamics of the equidistribution of points of small heights
  
  • Favre Charles
  • Bastien Fanny
  • Beaumont Vanille
  , 2013. Application to complex dynamics of the equidistribution of points of small heights
• Stabilization of an arbitrary profile for an ensemble of half-spin systems
  
  • Beauchard Karine
  • Pereira da Silva Paulo Sergio
  • Rouchon Pierre
  Automatica, Elsevier, 2013, 49 (7), pp.2133-2137. We consider the feedback stabilization of a variable profile for an ensemble of non interacting half-spins described by the Bloch equations. We propose an explicit feedback law that stabilizes asymptotically the system around a given arbitrary target profile. The convergence proof is done when the target profile is entirely in the south hemisphere or in the north hemisphere of the Bloch sphere. The convergence holds for initial conditions in a H1 neighbourhood of this target profile. This convergence is shown for the weak H1 topology. The proof relies on an adaptation of the LaSalle invariance principle to infinite dimensional systems. Numerical simulations illustrate the efficiency of these feedback laws, even for initial conditions far from the target profile. (10.1016/j.automatica.2013.03.011)
  DOI : 10.1016/j.automatica.2013.03.011
• Deux bases de données musicales avec développements musicxml : Carnet de Notes (PLM - analyse) et Neuma (CNRS - Bibliothèque numérique)
  
  • Tacaille Alice
  • Dang Nguyen-Bac
  , 2013. Conférence des Journées d'Informatique Musicale, MINT-OMF, 27 juin 2013
• Null controllability of Grushin-type operators in dimension two
  
  • Beauchard Karine
  • Cannarsa Piermarco
  • Guglielmi Roberto
  Journal of the European Mathematical Society, European Mathematical Society, 2014, 16 (1), pp.67-101. We study the null controllability of the parabolic equation associated with the Grushin-type operator in the rectangle, under an additive control supported in an open subset of the rectangle. We prove that the equation is null controllable in any positive time when the degeneracy is not too strong, and that there is no time for which it is null controllable when the degeneracy is too strong. In the transition regime and when the control support is a strip, a positive minimal time is required for null controllability. Our approach is based on the fact that, thanks to the particular geometric con guration, null controllability is closely linked to the one-dimensional observability of the Fourier components of the solution of the adjoint system, uniformly with respect to the Fourier frequency. (10.4171/JEMS/428)
  DOI : 10.4171/JEMS/428
• Endomorphism algebras of admissible $p$-adic representations of $p$-adic Lie groups
  
  • Dospinescu Gabriel
  • Schraen Benjamin
  Representation theory : An Electronic Journal of the American Mathematical Society, Amercian Mathematical Society, 2013, 17 (8), pp.237-246. Building on recent work of Ardakov and Wadsley, we prove Schur's lemma for absolutely irreducible admissible $p$-adic Banach space (respectively locally analytic) representations of $p$-adic Lie groups. We also prove finiteness results for the endomorphism algebra of an irreducible admissible representation. (10.1090/S1088-4165-2013-00432-6)
  DOI : 10.1090/S1088-4165-2013-00432-6
• Sharp Estimates for Turbulence in White-Forced Generalised Burgers Equation
  
  • Boritchev Alexandre
  , 2013. We consider a non-homogeneous generalised Burgers equation \frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - \nu \frac{\partial^2 u}{\partial x^2} = \eta,\ t \geq 0,\ x \in S^1. Here f is strongly convex and satisfies a growth condition, \nu is small and positive, while \eta is a random forcing term, smooth in space and white in time. For any solution u of this equation we consider the quasi-stationary regime, corresponding to t>=T_1, where T_1 depends only on f and on the distribution of \eta. We obtain sharp upper and lower bounds for Sobolev norms of $u$ averaged in time and in ensemble. These results yield sharp upper and lower bounds for natural analogues of quantities characterising the hydrodynamical turbulence. All our bounds do not depend on the initial condition or on t for t>=T_1, and hold uniformly in \nu. Estimates similar to some of our results have been obtained by Aurell, Frisch, Lutsko and Vergassola on a physical level of rigour; we use some arguments from their article.
• Construction of the Pauli-Villars-regulated Dirac vacuum in electromagnetic fields
  
  • Gravejat Philippe
  • Hainzl Christian
  • Lewin Mathieu
  • Séré Eric
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2013, 208 (2), pp.603-665. Using the Pauli-Villars regularization and arguments from convex analysis, we construct solutions to the classical time-independent Maxwell equations in Dirac's vacuum, in the presence of small external electromagnetic sources. The vacuum is not an empty space, but rather a quantum fluctuating medium which behaves as a nonlinear polarizable material. Its behavior is described by a Dirac equation involving infinitely many particles. The quantum corrections to the usual Maxwell equations are nonlinear and nonlocal. Even if photons are described by a purely classical electromagnetic field, the resulting vacuum polarization coincides to first order with that of full Quantum Electrodynamics. (10.1007/s00205-012-0609-1)
  DOI : 10.1007/s00205-012-0609-1