Nicolas LERNER

Institut de Mathématiques de Jussieu
175 rue du Chevaleret - 75013 Paris
Université Paris 6
France


Projet analyse fonctionnelle
Bureau 4B01
Téléphone: 01 44 27 85 70   (from abroad 331 44 27 85 70) lerner@math.jussieu.fr

A recent book

  • Metrics on the Phase Space and Non-Selfadjoint Pseudodifferential Operators, a book published by Birkhäuser in January 2010. This is a four-hundred-page book on the topic of pseudodifferential operators, with special emphasis on non-selfadjoint operators, a priori estimates and localization in the phase space.

    Recent papers

  • Instability of the Cauchy-Kovalevskaya solution for a class of non-linear systems, with Yoshinori Morimoto and Chao-Jiang Xu, American Journal of Mathematics, Vol. 132, 1, February 2010, pp. 99-123. We prove that in any C-infinity neighborhood of an analytic Cauchy datum, there exists a smooth function such that the corresponding initial value problem does not have any classical solution for a class of first-order non-linear systems. We use a method initiated by G. Métivier for elliptic systems based on the representation of solutions and on the FBI transform; in our case the system can be hyperbolic at initial time, but the characteristic roots leave the real line at positive times.

  • Fast rotating condensates in an asymmetric trap, with Amandine Aftalion and Xavier Blanc, article in press, Journal of Functional Analysis, Volume 257, Issue 3, 1 August 2009, Pages 753-806. We investigate the effect of the anisotropy of a harmonic trap on the behaviour of a fast rotating Bose - Einstein condensate. This is done in the framework of the 2D Gross - Pitaevskii equation and requires a symplectic reduction of the quadratic form defining the energy. This reduction allows us to simplify the energy on a Bargmann space and study the asymptotics of large rotational velocity. We characterize two regimes of velocity and anisotropy; in the first one where the behaviour is similar to the isotropic case, we construct an upper bound: a hexagonal Abrikosov lattice of vortices, with an inverted parabola profile. The second regime deals with very large velocities, a case in which we prove that the ground state does not display vortices in the bulk, with a 1D limiting problem. In that case, we show that the coarse grained atomic density behaves like an inverted parabola with large radius in the deconfined direction but keeps a fixed profile given by a Gaussian in the other direction. The features of this second regime appear as new phenomena.

  • Fast rotating condensates in an asymmetric harmonic trap, with Amandine Aftalion and Xavier Blanc, Physical Review A, Volume 79, Issue 1, Phys. Rev. A 79, 011603(R) (2009).

  • A note on the Oseen kernels, an article in Advances in Phase Space Analysis of Partial Differential Equations, PNLDE, vol. 78, Birkhäuser, 2009. We give an explicit expression for the kernels of the Oseen operators, Δ-1 ∂xj ∂xk etΔ. These Fourier multipliers involve the incomplete gamma function and the confluent hypergeometric functions of the first kind. This explicit expression provides directly the classical decay estimates with sharp bounds.

  • Semi-classical estimates for non-selfadjoint operators, The Asian Journal of Mathematics, vol.11, 2, 217-250, (2007). pdf. This is a survey paper on the topic of proving or disproving a priori L2 estimates for non-selfadjoint operators. Our framework will be limited to the case of scalar semi-classical pseudodifferential operators of principal type. We start with recalling the simple conditions following from the sign of the first bracket of the real and imaginary part of the principal symbol. Then we introduce the geometric condition (ψ) and show the necessity of that condition for obtaining a weak L2 estimate. Considering that condition satisfied, we investigate the finite-type case, where one iterated bracket of the real and imaginary part does not vanish, a model of subelliptic operators. The last section is devoted partly to rather recent results, although we begin with a version of the 1973 theorem of R.Beals and C.Fefferman on solvability with loss of one derivative under condition (P); next, we present a 1994 counterexample by N.L. establishing that condition (ψ) does not ensure an estimate with loss of one derivative for P*. Finally, we show that condition (ψ) implies an estimate with loss of 3/2 derivatives, following the recent papers by N.Dencker and N.L.

  • On the Fefferman-Phong inequality and a Wiener-type algebra of pseudodifferential operators, with Yoshinori Morimoto, Publications of the Research Institute for Mathematical Sciences (Kyoto University) 43, 329-371, (2007), pdf. We provide an extension of the Fefferman-Phong inequality to nonnegative symbols whose fourth derivative belongs to a Wiener-type algebra of pseudodifferential operators introduced by J.Sjöstrand. As a byproduct, we obtain that the number of derivatives needed to get the classical Fefferman-Phong inequality in D dimensions is bounded above by 2D+4+ε. Our method relies on some refinements of the Wick calculus, which is closely linked to Gabor wavelets. Also we use a decomposition of C3,1 nonnegative functions as a sum of squares of C1,1 functions with sharp estimates. In particular, we prove that a C3,1 nonnegative function can be written as a finite sum Σ bj2, where each bj is C1,1, but also where each function bj2 is C3,1. A key point in our proof is to give some bounds on (bj'bj'')' and on (bjbj'')''.

  • Cutting the loss of derivatives for solvability under condition (Ψ), Bulletin de la Société Mathématique de France, vol.134, 4, 559-631, (2006). For a principal type pseudodifferential operator, we prove that condition (Ψ) implies local solvability with a loss of 3/2 derivatives. We use many elements of Dencker's paper on the proof of the Nirenberg-Treves conjecture and we provide some improvements of the key energy estimates which allows us to cut the loss of derivatives from 2 (Dencker's result) to 3/2 (the present paper). It is already known that condition (Ψ) does not imply local solvability with a loss of 1 derivative, so we have to content ourselves with a loss >1. Since this paper is quite technical, it could be a good idea to begin with the transparencies of my talk at the Bourbaki seminar in March 2006. A more detailed presentation pdf appeared in the proceedings of that seminar (Astérisque, vol.311, exposé 960, (2007)).

  • Transport equations with partially BV velocities, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Serie V, Vol. III, fasc.4 (2004), pdf, dvi. In this article, we prove the uniqueness of weak solutions for a class of transport equations whose velocities are partially with bounded variation. Our result deals with the vector field
    X = a1(x1) .∂x1 + a2(x1,x2) .∂x2
    where a1(x1) is a BV function and a2(x1,x2) is only L1 with respect to x1 and BV with respect to x2, with a boundedness condition on the divergence of each vector field a1, a2. This model was studied in a recent paper by P.-L.Lions and C.Le Bris with a W1,1 regularity assumption replacing our BV hypothesis. This settles partly a question raised in a forthcoming paper by L.Ambrosio. We examine the details of the argument of that article and we combine some consequences of the Alberti rank-one structure theorem for BV vector fields with a regularization procedure. Our regularization kernel is not restricted to be a convolution and is introduced as an unknown function. Our method amounts to commute a pseudo-differential operator with a BV function.

  • Équations de transport dont les vitesses sont partiellement BV, texte de l'exposé du 20 janvier 2004 au séminaire X-EDP , pdf. Essentially a french version of the above article. However, we also go back to vector fields X as above with W1,1 regularity: in that case, our boundedness condition on the divergence is only on the whole div X and not on each divergence of a1, a2.
  • Uniqueness of L∞ solutions for a class of conormal BV vector fields, with Ferruccio Colombini, pdf, dvi, an article in Geometric Analysis of PDE and Several Complex Variables,(editors S. Chanillo, P. Cordaro, N. Hanges, J. Hounie, and A. Meziani) Contemporary Mathematics #368. In this paper, we prove the uniqueness of bounded measurable solutions for a class of vector fields with bounded variation. Our class contains the piecewise W1,1 class. We use some arguments of geometric measure theory to get rid of sets whose d-1 Hausdorff measure is 0. Also we need an anisotropic regularization argument.

  • Une procédure de Calderón-Zygmund pour le problème de la racine k-ième, avec Ferruccio Colombini, Annali di Matematica Pura ed Applicata, volume 182, 231-246, 2003, pdf.

  • The Wick calculus of pseudo-differential operators and some of its applications, in the Chilean journal CUBO, volume 5, (1), 2003. pdf.

  • Solving pseudo-differential equations, pdf, dvi, my article published in the Proceedings of the ICM 2002 in Beijing, Higher Education Press, Volume II, pages 711-720.

  • Uniqueness of continuous solutions for BV vector fields, with Ferruccio Colombini, Duke Mathematical Journal, volume 111, No.2, pages 357-384, 2002.

  • On the existence and uniqueness of solutions to stochastic equations in infinite dimension with integral-Lipschitz coefficients, with Ying Hu, Journal of Mathematics of Kyoto University, volume 42, (3), pages 579 - 598, 2002, pdf.

    A tribute to Laurent Schwartz. Hommage à Laurent Schwartz, pdf. This text corresponds to a conference celebrating the memory of Laurent Schwartz, given in november 2003 at the University of Rennes.

    Lectures on Integration. Cours d'Intégration en ligne.

    Lecture Notes on Real Analysis.

    M2, Analyse de Fourier, cours introductif.

    M2, Inégalités classiques en analyse harmonique, cours fondamental.

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