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Yannick Sire

  1. Fast iteration of cocyles over rotations and Computation of hyperbolic bundles.

    Втр, 02/15/2011 - 16:01 — SomNambul
    Authors: Yannick Sire, Rafael de la Llave, Gemma Huguet
    Subjects: Dynamical Systems
    Abstract

    In this paper, we develop numerical algorithms that use small requirements of
    storage and operations for the computation of hyperbolic cocycles over a
    rotation. We present fast algorithms for the iteration of the quasi-periodic
    cocycles and the computation of the invariant bundles, which is a preliminary
    step for the computation of invariant whiskered tori.

  2. Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates.

    Втр, 12/07/2010 - 16:01 — SomNambul
    Authors: Yannick Sire, Xavier Cabre
    Subjects: Analysis of PDEs
    Abstract

    This is the first of two articles dealing with the equation $(-\Delta)^{s} v=
    f(v)$ in $\mathbb{R}^{n}$, with $s\in (0,1)$, where $(-\Delta)^{s}$ stands for
    the fractional Laplacian ---the infinitesimal generator of a L\'evy process.
    This equation can be realized as a local linear degenerate elliptic equation in
    $\mathbb{R}^{n+1}_+$ together with a nonlinear Neumann boundary condition on
    $\partial \mathbb{R}^{n+1}_+=\mathbb{R}^{n}$.

  3. Computation of whiskered invariant tori and their associated manifolds: new fast algorithms.

    Пт, 04/30/2010 - 15:01 — SomNambul
    Authors: Yannick Sire, Rafael de la Llave, Gemma Huguet
    Subjects: Dynamical Systems
    Abstract

    In this paper we present efficient algorithms for the computation of several
    invariant objects for Hamiltonian dynamics. More precisely, we consider KAM
    tori (i.e diffeomorphic copies of the torus such that the motion on them is
    conjugated to a rigid rotation) both Lagrangian tori (of maximal dimension) and
    whiskered tori (i.e. tori with hyperbolic directions which, together with the
    tangents to the torus and the symplectic conjugates span the whole tangent
    space).

  4. A note on precised Hardy inequalities on Carnot groups and Riemannian manifolds.

    Пнд, 03/29/2010 - 15:01 — SomNambul
    Authors: Yannick Sire, Emmanuel Russ
    Subjects: Functional Analysis
    Abstract

    We prove non local Hardy inequalities on Carnot groups and Riemannian
    manifolds, relying on integral representations of fractional Sobolev norms.

  5. Some possibly degenerate elliptic problems with measure data and non linearity on the boundary.

    Пнд, 03/01/2010 - 16:01 — SomNambul
    Authors: Yannick Sire, Thierry Gallouët
    Subjects: Analysis of PDEs
    Abstract

    The goal of this paper is to study some possibly degenerate elliptic equation
    in a bounded domain with a nonlinear boundary condition involving measure data.
    We investigate two types of problems: the first one deals with the laplacian in
    a bounded domain with measure supported on the domain and on the boundary. A
    second one deals with the same type of data but involves a degenerate weight in
    the equation. In both cases, the nonlinearity under consideration lies on the
    boundary.

  6. Non local Poincar\'e inequalities on Lie groups with polynomial volume growth.

    Пнд, 01/25/2010 - 16:01 — SomNambul
    Authors: Yannick Sire, Emmanuel Russ
    Subjects: Functional Analysis
    Abstract

    Let $G$ be a real connected Lie group with polynomial volume growth, endowed
    with its Haar measure $dx$. Given a $C^2$ positive function $M$ on $G$, we give
    a sufficient condition for an $L^2$ Poincar\'e inequality with respect to the
    measure $M(x)dx$ to hold on $G$. We then establish a non-local Poincar\'e
    inequality on $G$ with respect to $M(x)dx$.

  7. Fractional Poincar\'e inequalities for general measures.

    Ср, 11/25/2009 - 16:01 — SomNambul
    Authors: Clément Mouhot, Yannick Sire, Emmanuel Russ
    Subjects: Analysis of PDEs
    Abstract

    We prove a fractional version of Poincar\'e inequalities in the context of
    $\R^n$ endowed with a fairly general measure. Namely we prove a control of an
    $L^2$ norm by a non local quantity, which plays the role of the gradient in the
    standard Poincar\'e inequality. The assumption on the measure is the fact that
    it satisfies the classical Poincar\'e inequality, so that our result is an
    improvement of the latter inequality. Moreover we also quantify the tightness
    at infinity provided by the control on the fractional derivative in terms of a
    weight growing at infinity.

  8. Singular solutions of fractional order conformal Laplacians.

    Пт, 11/06/2009 - 16:01 — SomNambul
    Authors: Rafe Mazzeo, Yannick Sire, Maria del Mar Gonzalez
    Subjects: Differential Geometry
    Abstract

    We investigate the singular sets of solutions of conformally covariant
    elliptic operators of fractional order with the goal of developing
    generalizations of some well-known properties of solutions of the singular
    Yamabe problem.

  9. A Liouville theorem for non local elliptic equations.

    Чт, 09/10/2009 - 10:12 — SomNambul
    Authors: Louis Dupaigne, Yannick Sire
    Subjects: Analysis of PDEs
    Abstract

    We prove a Liouville-type theorem for bounded stable solutions $v \in
    C^2(\R^n)$ of elliptic equations of the type (-\Delta)^s v= f(v)\qquad {in
    $\R^n$,} where $s \in (0,1)$ {and $f$ is any nonnegative function}. The
    operator $(-\Delta)^s$ stands for the fractional Laplacian, a
    pseudo-differential operator of symbol $|\xi |^{2s}$.

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