Publications

2006

• Modifications et cycles proches sur une base générale
  
  • Orgogozo Fabrice
  International Mathematics Research Notices, Oxford University Press (OUP), 2006, 2006, pp.Article ID 25315, 38 pages. Si l'on étudie les cycles évanescents, c'est-à-dire la cohomologie (étale) des fibres de Milnor d'un morphisme de but de dimension >1, on perd les propriétés, démontrées par P. Deligne dans SGA 4 1/2, de commutation au changement de base et constructibliité. Dans cet article, on montre, après avoir rappelé la définition du complexe des cycles proches dans ce contexte, qu'on retrouve ces propriétés après modification de la base. L'ingrédient essentiel est un théorème de A.J. de Jong sur les fibrations plurinodales. Une application du formalisme aux pinceaux de Lefschetz est donnée. (10.1155/IMRN/2006/25315)
  DOI : 10.1155/IMRN/2006/25315
• Construction of Sobolev spaces of fractional order with sub-riemannian vector fields
  
  • Vigneron François
  • Mustapha Sami
  , 2006.
• Multiple solutions to the supercritical Bahri-Coron's problem in pierced domains
  
  • Pistoia Angela
  • Rey Olivier
  Advances in Differential Equations, Khayyam Publishing, 2006, 11 (6), pp.647-666. We consider the supercritical Dirichlet problem $$\left(P_\epsilon\right)\qquad -\Delta u=u^{{N+2\over N-2}+\epsilon}\ \hbox{in $\Omega$},\ u>0\ \hbox{in $\Omega,$}\ u=0\ \hbox{on $\partial\Omega$} $$ where $N\ge3,$ $\epsilon>0$ and $\Omega\subset\mathbb{R}^N$ is a smooth bounded domain with a small hole of radius $d.$ When $\Omega$ has some symmetries, we show that $\left(P_\epsilon\right)$ has an arbitrary number of solutions for $\epsilon$ and $d$ small enough. When $\Omega$ has no symmetries, we prove the existence, for $d$ small enough, of solutions blowing up at two or three points close to the hole as $\epsilon$ goes to zero.
• Mirror symmetry for weighted projective planes and their noncommutative deformations
  
  • Auroux Denis
  • Katzarkov L.
  • Orlov Dmitri
  , 2006.
• The canonical pencils on Hokirawa surfaces
  
  • Auroux Denis
  , 2006.
• On the localization of the magnetic and the velocity fields in the equations of magnetohydrodynamics
  
  • Brandolese Lorenzo
  • Vigneron François
  Proceedings of the Royal Society of Edinburgh: Section A, Mathematics, Royal Society of Edinburgh, 2006, 2007 (137), pp.475--495. We study the behavior at infinity, with respect to the space variable, of solutions to the magnetohydrodynamics equations in ${\bf R}^d$. We prove that if the initial magnetic field decays sufficiently fast, then the plasma flow behaves as a solution of the free nonstationnary Navier--Stokes equations when $|x|\to +\infty$, and that the magnetic field will govern the decay of the plasma, if it is poorly localized at the beginning of the evolution. Our main tools are boundedness criteria for convolution operators in weighted spaces.
• Hamiltonian 2-forms in Kahler geomtry
  
  • Gauduchon Paul
  Journal of Differential Geometry, International Press, 2006, 73 N°3, pp.359-412.
• An anomaly formula for L2-analytic torsions on manifolds with boundary
  
  • Ma Xiaonan
  • Zhang Weiping
  , 2006, pp.247-274.
• The odd-dimensional Goldberg Conjecture
  
  • Apostolov Vestislav
  • Draghici Tedi
  • Moroianu Andrei
  Mathematical News / Mathematische Nachrichten, Wiley-VCH Verlag, 2006, 279, pp.948-952. An odd-dimensional version of the Goldberg conjecture was formulated and proved by Boyer and Galicki, using an orbifold analogue of Sekigawa's formulas, and an approximation argument of K-contact structures with quasi-regular ones. We provide here another proof of this result and give some applications. (10.1002/mana.200410404)
  DOI : 10.1002/mana.200410404
• On exponential observability estimates for the heat semigroup with explicit rates
  
  • Miller Luc
  Rendiconti Lincei. Matematica e Applicazioni, European Mathematical Society, 2006, 17 (4), pp.351--366. This note concerns the final time observability inequality from an interior region for the heat semigroup, which is equivalent to the null-controllability of the heat equation by a square integrable source supported in this region. It focuses on exponential estimates in short times of the observability cost, also known as the control cost and the minimal energy function. It proves that this final time observability inequality implies four variants (an integrated inequality with singular weights, an integrated inequality in infinite times, a sharper inequality and a Sobolev inequality) with roughly the same exponential rate everywhere and some control cost estimates with explicit exponential rates concerning null-controllability, null-reachability and approximate controllability. A conjecture and open problems about the optimal rate are stated. This note also contains a brief review of recent or to be published papers related to exponential observability estimates: boundary observability, Schrödinger group, anomalous diffusion, thermoelastic plates, plates with square root damping and other elastic systems with structural damping.
• Hamiltonian 2-forms in Kähler gerometry I : general theory
  
  • Apostolov Vestislav
  • Calderbank David M. J.
  • Gauduchon Paul
  Journal of Differential Geometry, International Press, 2006, 73 (3), pp.359-412.
• On the asymptotic expansion of Bergman kernel
  
  • Ma Xiaonan
  • Liu Kefeng
  • Dai Xianzhe
  Journal of Differential Geometry, International Press, 2006, 72, pp.1-41.
• Mapping class group factorizations and symplectic 4-manifolds: some open problems
  
  • Auroux Denis
  , 2006, pp.123--132.
• An anomaly formula for Ray-Singer metrics on manifolds with boundary
  
  • Ma Xiaonan
  • Bruning J.
  Geometric And Functional Analysis, Springer Verlag, 2006, 16 n°4, pp.767-837.
• Killing Forms on Symmetric Spaces
  
  • Belgun Florin
  • Moroianu Andrei
  • Semmelmann Uwe
  Differential Geometry and its Applications, Elsevier, 2006, 24, pp.215-222. Killing forms on Riemannian manifolds are differential forms whose covariant derivative is totally skew--symmetric. We show that a compact simply connected symmetric space carries a non--parallel Killing $p$--form ($p\ge2$) if and only if it isometric to a Riemannian product $S^k\times N$, where $S^k$ is a round sphere and $k>p$. (10.1016/j.difgeo.2005.09.007)
  DOI : 10.1016/j.difgeo.2005.09.007
• Non-structural controllability of linear elastic systems with structural damping
  
  • Miller Luc
  Journal of Functional Analysis, Elsevier, 2006, 236 (2), pp.592-608. This paper proves that any initial condition in the energy space for the plate equation with square root damping z''- r Delta z' + Delta^2 z' = u on a smooth bounded domain, with hinged boundary conditions z=Delta z=0, can be steered to zero by a square integrable input function u supported in arbitrarily small time interval [0,T] and subdomain. As T tends to zero, for initial states with unit energy norm, the norm of this u grows at most like exp(C_p /T^p) for any real p>1 and some C_p>0. Indeed, this fast controllability cost estimate is proved for more general linear elastic systems with structural damping and non-structural controls satisfying a spectral observability condition. Moreover, under some geometric optics condition on the subdomain allowing to apply the control transmutation method, this estimate is improved into p=1 and the dependence of C_p on the subdomain is made explicit. These results are analogous to the optimal ones known for the heat flow. (10.1016/j.jfa.2006.03.001)
  DOI : 10.1016/j.jfa.2006.03.001
• The control transmutation method and the cost of fast controls
  
  • Miller Luc
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2006, 45 (2), pp.762-772. In this paper, the null controllability in any positive time T of the first-order equation (1) x'(t)=e^{i\theta}Ax(t)+Bu(t) (|\theta|<\pi/2 fixed) is deduced from the null controllability in some positive time L of the second-order equation (2) z''(t)=Az(t)+Bv(t). The differential equations (1) and (2) are set in a Banach space, B is an admissible unbounded control operator, and A is a generator of cosine operator function. The control transmutation method explicits the input function u of (1) in terms of the input function v of (2): u(t,x)=\int k(t,s)v(s)ds, where the compactly supported kernel k depends on T and L only. It proves that the norm of a u steering the system (1) from an initial state x_{0} to zero grows at most like ||x_{0}||\exp(\alpha_{*}L^{2}/T) as the control time T tends to zero. (The rate \alpha_{*} is characterized independently by a one-dimensional controllability problem.) In the applications to the cost of fast controls for the heat equation, L is the length of the longest ray of geometric optics which does not intersect the control region. (10.1137/S0363012904440654)
  DOI : 10.1137/S0363012904440654
• New Asymptotic profiles of nonstationnary solutions of the Navier-Stokes systems
  
  • Vigneron François
  , 2006.
• Determining a magnetic Schrodinger operator from partial Cauchy data
  
  • Sjostrand Johannes
  • dos Santos Ferreira David
  • Kenig Carlos
  • Uhlmann Gunther
  , 2006.
• Groupes et Symetries: groupes finis et algèbres de Lie, représentations, deuxième édition révisée
  
  • Kosmann-Schwarzbach Yvette
  , 2006, pp.193.