Application lipschitzienne | EPFL Graph Search

En analyse mathématique, une application lipschitzienne (du nom de Rudolf Lipschitz) est une application possédant une certaine propriété de régularité qui est plus forte que la continuité. Intuitivement, c'est une fonction qui est limitée dans sa manière d'évoluer. Tout segment reliant deux points du graphe d'une telle fonction aura une pente inférieure, en valeur absolue, à une constante appelée constante de Lipschitz. Les fonctions lipschitziennes sont un cas particulier de fonctions höldériennes. Définitions Cas réel thumb|upright=1.8|alt=Illustation graphique avec un cône|Une fonction réelle est k-lipschitzienne si le double cône blanc peut se déplacer le long de son graphe sans que jamais la courbe de la fonction passe à l'intérieur. Plus k est petit, plus le cône blanc s'élargit et moins la fonction peut être abrupte. Soient E une partie de ℝ, f:E\to\R une application et k un réel positif. On dit que f est k-lipschitzienne si \forall

On long time behavior of solutions to nonlinear dispersive equations

Stefano Francesco Burzio

In part I, we address the issue of existence of solutions for Cauchy problems involving nonlinear hyperbolic equations for initial data in Sobolev spaces with scaling subcritical regularity. In particular, we analyse nonlinear estimates for null-forms in the context of wave Sobolev spaces

H^{s,b}

, first in a flat background, then we generalize to more general curved backgrounds. We provide the foundations to show that the Yang-Mills equation in

\mathbb{R}^{1+3}

are globally well-posedness for small weighted

H^{3/4+}\times H^{-1/4+}

initial data, matching the minimal regularity obtained by Tao \cite{tao03}. Our method, inspired from \cite{dasgupta15}, combines the classical Penrose compactification of Minkowski space-time with a null-form estimates for second order hyperbolic operators with variable coefficients. The proof of the null-form appearing in the Yang-Mills equation will be provided in a subsequent work. As a consequence of our argument, we shall obtain sharp pointwise decay bounds. In part II, we show that the finite time type II blow-up solutions for the energy critical nonlinear wave equation

\Box u = -u^5

\R^{3+1}

constructed in \cite{krieger2009slow}, \cite{krieger2009renormalization} are stable along a co-dimension one Lipschitz manifold of data perturbations in a suitable topology, provided the scaling parameter

\lambda(t) = t^{-1-\nu}

is sufficiently close to the self-similar rate, i. e.

\nu>0

is sufficiently small. This result is qualitatively optimal in light of the result of \cite{krieger2015center}, it builds on the analysis of \cite{krieger2017stability} and it is joint work with my thesis advisor Prof. J. Krieger.

EPFL2020

An Accelerated First-Order Method for Non-convex Optimization on Manifolds

Nicolas Boumal, Christopher Arnold Criscitiello

We describe the first gradient methods on Riemannian manifolds to achieve accelerated rates in the non-convex case. Under Lipschitz assumptions on the Riemannian gradient and Hessian of the cost function, these methods find approximate first-order critical points faster than regular gradient descent. A randomized version also finds approximate second-order critical points. Both the algorithms and their analyses build extensively on existing work in the Euclidean case. The basic operation consists in running the Euclidean accelerated gradient descent method (appropriately safe-guarded against non-convexity) in the current tangent space, then moving back to the manifold and repeating. This requires lifting the cost function from the manifold to the tangent space, which can be done for example through the Riemannian exponential map. For this approach to succeed, the lifted cost function (called the pullback) must retain certain Lipschitz properties. As a contribution of independent interest, we prove precise claims to that effect, with explicit constants. Those claims are affected by the Riemannian curvature of the manifold, which in turn affects the worst-case complexity bounds for our optimization algorithms.

SPRINGER2022

Equations de type implicite avec contraintes

Chiara Tanteri

In this thesis we will treat the Dirichlet problem for systems of implicit equations, i.e. where Ω ⊂ Rn is an open set, u : Ω → Rm, Fi : Ω × Rm × Rm×n → R, m,n ≥ 1, are continuous functions and φ, the boundary datum, is given. At first we will be interested in the study of problems of the type (1) under constraints. We will show theorems of existence of Lipschitz solutions using an approach based on Baire category theorem. As corollaries of our abstract results, we will give two theorems related to the constraints det Du > 0 and det Du = 1, the constraints most closely related to the applications. Indeed, the constraints det Du > 0 and det Du = 1 come from nonlinear elasticity and represent respectively the conditions of non interpenetration of matter and incompressibility. In the second part of this study, we will focus on the applications. We will treat many examples such as the case of singular values, potential wells under the incompressibility constraint (i.e. det Du = l), the problem of confocal ellipses, the problem of nematic elastomers, particularly related to the fields of nonlinear elasticity, the microstructure of the crystals and the optimal design, as well as the complex eikonal equation (application related to geometric optics). From the mathematical point of view, we will give sufficient conditions of solvability of the system (1). These conditions consist in characterizing the different convex hulls. Indeed, the possibility of representing these sets, in algebraic terms, gives one of the conditions which the boundary datum must satisfy so that a problem of the type (1) admits a solution. Finally we will extend to polyconvex sets properties such as the gauge, the characterization of the extreme points and the Choquet function, all of which are well-known tools within the framework of classical convex analysis.

EPFL2000