Publications

1999

• Mathematical analysis of elastic surface waves in topographic waveguides.
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Duterte Jean
  • Joly Patrick
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 1999, 9 (5), pp.755-798. We present here a theoretical study of the guided waves in an isotropic homogeneous elastic half-space whose free surface has been deformed. The deformation is supposed to be invariant in the propagation direction and localized in the transverse ones. We show that finding guided waves amounts to solving a family of 2-D eigenvalue problems set in the cross-section of the propagation medium. Then using the min-max principle for non-compact self-adjoint operators, we prove the existence of guided waves for some particular geometries of the free surface. These waves have a smaller speed than that of the Rayleigh wave in the perfect half-space and a finite transverse energy. Moreover, we prove that the existence results are valid for arbitrary high frequencies in the presence of singularities of the free boundary. Finally, we prove that no guided mode can exist at low frequency, except maybe the fundamental one. (10.1142/S0218202599000373)
  DOI : 10.1142/S0218202599000373
• Thermodynamics of a two-dimensionnal unbounded self-gravitating system
  
  • Perez Jérôme
  • Aly Jean-Jacques
  Physical Review E : Statistical, Nonlinear, and Soft Matter Physics [2001-2015], American Physical Society, 1999, 60, pp.5185. The thermodynamics of a two-dimensional self-gravitating system occupying the whole plane is considered in the mean-field approximation. First, it is proven that, if the number N of particles and the total energy E are imposed as the only external constraints, then the entropy admits the least upper bound S+(N,E)=2E/N+N ln(eπ2) (in appropriate units). Moreover, there does exist a unique state of maximum entropy, which is characterized by a Maxwellian distribution function with a temperature T=N/2 independent of E. Next, it is shown that, if the total angular momentum J is imposed as a further constraint, the largest possible value of the entropy does not change, and there is no admissible state of maximum entropy, but in the case J=0. Finally, some inequalities satisfied by a class of so-called H functions and related generalized entropies are given. (10.1103/PhysRevE.60.5185)
  DOI : 10.1103/PhysRevE.60.5185
• Caractérisation de la partie singulière et résolution des équations de Maxwell en géométrie singulière axisymétrique
  
  • Assous Franck
  • Ciarlet Patrick
  • Labrunie Simon
  Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 1999, 328 (9), pp.767-772. On étudie les équations de Maxwell stationnaires dans un ouvert Ω non régulier, non convexe, à symétrie axiale. L'espace des solutions s'écrit comme la somme orthogonale d'une partie régulière, contenue dans H1(Ω)3 et d'une partie singulière. On montre que, comme dans le cas bidimensionnel, la partie singulière est reliée aux fonctions propres singulières (axisymétriques) du laplacien, et est de dimension finie. (10.1016/S0764-4442(99)80269-9)
  DOI : 10.1016/S0764-4442(99)80269-9
• Fully discrete finite element approaches for time-dependent Maxwell's equations
  
  • Ciarlet Patrick
  • Zou Jun
  Numerische Mathematik, Springer Verlag, 1999, 82 (2), pp.193-219. A fully discrete finite element method is used to approximate the electric field equation derived from time-dependent Maxwell's equations in three dimensional polyhedral domains. Optimal energy-norm error estimates are achieved for general Lipschitz polyhedral domains. Optimal L2 -norm error estimates are obtained for convex polyhedral domains. (10.1007/s002110050417)
  DOI : 10.1007/s002110050417
• Tools for solving the div-curl problem with mixed boundary conditions in a polygonal domain
  
  • Ciarlet Patrick
  , 1999. Following [Assous, Ciarlet, Jr., Sonnendrücker, Resolution of the Maxwell equations in a domain with reentrant corners (1998)], we continue to study the resolution of two-dimensional problems in non convex domains. In this previous paper, we considered several methods for solving Maxwell’s equations (stationary or instationary) with a perfectly conducting boundary condition.
• The Inverse EEG and MEG Problems : The Adjoint State Approach I: The Continuous Case
  
  • Faugeras Olivier
  • Clément François
  • Deriche Rachid
  • Keriven Renaud
  • Papadopoulo Théodore
  • Roberts Jean
  • Viéville Thierry
  • Devernay Frédéric
  • Gomes José
  • Hermosillo Gerardo
  • Kornprobst Pierre
  • Lingrand Diane
  , 1999, pp.28. In this report, we study the problem of the three-dimensional reconstruction of the electrical activity of the brain from electroencephalography (EEG) and magnetoencephalography (MEG). We use a variational approach based upon three main methods and ideas. The first one is the optimal control of systems governed by elliptic partial differential equations, the second is the regularization of the solutions while preserving the discontinuities (the edges), and the third one is the use of geometric information obtained from magnetic resonance images (MRI) to constrain the solutions in an anatomically «reasonable» way.
• Stability of rotating spherical stellar systems
  
  • Alimi J.-M.
  • Perez Jérôme
  • Serna A.
  Monthly Notices of the Royal Astronomical Society, Oxford University Press (OUP): Policy P - Oxford Open Option A, 1999, 305 (4), pp.859-865. We study the stability of rotating collisionless self-gravitating spherical systems by using highresolution N-body experiments on a Connection Machine CM-5.We added rotation to Ossipkov±Merritt (OM) anisotropic spherical systems by using twomethods. The first method conserves the anisotropy of the distribution function defined in the OM algorithm. The second method distorts the systems in velocity-space. We then show that the stability of systems depends both on their anisotropy and on the value of the ratio of the total kinetic energy to the rotational kinetic energy. We also test the relevance of the stability parameters introduced by Perez et al. for the case of rotating systems (10.1046/j.1365-8711.1999.02475.x)
  DOI : 10.1046/j.1365-8711.1999.02475.x
• A singular field method for the solution of Maxwell's equations
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Hazard Christophe
  • Lohrengel Stéphanie
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 1999, 59 (6), pp.2028-2044. It is well known that in the case of a regular domain the solution of the time-harmonic Maxwell's equations allows a discretization by means of nodal finite elements: this is achieved by solving a regularized problem similar to the vector Helmholtz equation. The present paper deals with the same problem in the case of a nonconvex polyhedron. It is shown that a nodal finite element method does not approximate in general the solution to Maxwell's equations, but actually the solution to a neighboring variational problem involving a different function space. Indeed, the solution to Maxwell's equations presents singularities near the edges and corners of the domain that cannot be approximated by Lagrange finite elements. A new method is proposed involving the decomposition of the solution field into a regular part that can be treated numerically by nodal finite elements and a singular part that has to be taken into account explicitly. This singular field method is presented in various situations such as electric and magnetic boundary conditions, inhomogeneous media, and regions with screens. Copyright © 1999 Society for Industrial and Applied Mathematics (10.1137/S0036139997323383)
  DOI : 10.1137/S0036139997323383
• A Hamilton-Jacobi equation with measures arising in Gamma-convergence of optimal control problems
  
  • Briani Ariela
  Differential and integral equations, Khayyam Publishing, 1999, 12 (6), pp.849-886. We consider a Hamilton-Jacobi equation with a measure in the Hamiltonian arising from the $\Gamma$-limit of some optimal control problems. We give a definition of viscosity solution in this case, by adapting the method of the reparametrization of Dal Maso and Rampazzo
• Mixed Finite Elements, Strong Symmetry and Mass Lumping for Elastic Waves
  
  • Bécache Eliane
  • Joly Patrick
  • Tsogka Chrysoula
  , 1999. We present here the continuation of our work on mixed finite elements for wave propagation problems. In a previous report, we constructed and analysed a new family of quadrangular (2D) or cubic (3D) mixed finite elements, for the approximation of the scalar anisotropic wave equation. This work is extended here to the elastic wave equation, including in the case of an anisotropic medium. These new elements present the specificity to enforce the symmetry of the stress tensor in a str ong way and lead to explicit schemes (via mass lumping), after time discretization. The convergence analysis of these mixed finite elements is not straightforward: neither the standard abstract theory nor the theory we developed for the scalar case can be applied. That is why we introduce a new abstract theory which allows to get error estimates.
• On the minimization of the energy of a free-electron gas with constrained density function
  
  • Bokanowski Olivier
  • Schindler Ian
  • Zidani Hasnaa
  Nonlinear Analysis: Theory, Methods and Applications, Elsevier, 1999, 35 (8), pp.1073-1090. One of the aims of density functional theory is to obtain properties of (ground) states of large systems, in particular their energy, by solving a nonlinear equation, involving only the parameters of a single electron. The oldest such theory is the Thomas-Fermi approach. Major developments are due to Hohenberg, Kohn and Sham in the mid-1960s and the present article discusses an approximate nonlinear equation arising within the context of the Kohn-Sham approach [P. Hohenberg and W. Kohn, Phys. Rev. (2) 136 (1964), B864-B871; MR0180312 (31 #4547); W. Kohn and L. J. Sham, Phys. Rev. (2) 140 (1965), A1133-A1138; MR0189732 (32 #7154)]. It is connected to the exchange contribution to the Kohn-Sham functional. The latter arises from taking into account the Fermi-Dirac statistics and complicates the application of a variational procedure significantly. Thus it is often replaced by an approximate term and the authors discuss the existence of a solution for this situation. They make a few physically reasonable assumptions and are then able to demonstrate the existence of a proper solution.
• Analyse spectrale et singularités d'un problème de transmission non coercif
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Dauge Monique
  • Ramdani Karim
  Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 1999, 328 (8), pp.717-720. Cette Note est consacrée à l'analyse spectrale d'un opérateur non borné associé à un problème de transmission bidimensionnel non coercif. Nous montrons par une méthode d'équations intégrales que si l'interface est régulière, cet opérateur est auto-adjoint à résolvante compacte. Si l'interface présente un coin, une étude des singularités par transformée de Mellin permet d'obtenir une condition nécessaire et suffisante, portant sur le contraste entre les deux milieux, pour que l'opérateur soit auto-adjoint. S'il ne l'est pas, nous donnons une caractérisation de ses extensions auto-adjointes. (10.1016/S0764-4442(99)80241-9)
  DOI : 10.1016/S0764-4442(99)80241-9
• A Eulerian Method for Capturing Caustics
  
  • Benamou Jean-David
  • Solliec Ian
  , 1999. A robust numerical method for the localization of caustics is proposed for general Hamiltonians. It is based on the direct resolution of a system of partial differential equations obtained through a local change of the time variable in the Hamilton-Jacobi equation and complemented by a set of transport equation. Numerical results (1 to 3-D) are presented.
• Schémas d'ordre élevé en espace et/ou en temps pour l'équation des ondes acoustiques 1-D
  
  • Anné Laurent
  • Joly Patrick
  • Tran Quang Huy
  , 1999. Dans cet article, nous nous proposons de : -Etablir toutes les propriétés, déjà signalées en conjecture [2] ou jusque-là méconnues, concernant la discrétisation à un ordre élevé du laplacien 1D. - Généraliser aux ordres supérieurs le schéma aux différences finies 4-4 de Cohen [6] pour l'équation des ondes acoustiques en milieu homogène 1-D. Pour la famille de schémas 2m-2m obtenue, des propriétes variées concernant la stabilité et la dispersion numérique sont démentrées. La preuve de cse résultats se fait à l'aide de mathématiques élémentaires.
• High Frequency Limit of the Helmholtz Equations
  
  • Benamou Jean-David
  • Castella François
  • Katsaounis Thodoros
  • Perthame Benoît
  , 1999. We derive the high frequency limit of the Helmholtz equations in terms of quadratic observables. We prove that it can be written as a stationary Liouville equation with source terms. Our method is based on the Wigner Transform, which is a classical tool for evolution dispersive equations. We extend its use to the stationary case after an appropriate scaling of the Helmholtz equation. Several specific difficulties arise here; first, the identification of the source term (which does not share the quadratic aspect) in the limit, then, the lack of L2 bounds which can be handled with homogeneous Morrey-Campanato estimates, and finally the problem of uniqueness which, at several stage of the proof, is related to outgoing conditions at infinity.