Groupe de travail Analyse

Consider a control system 𝛛t f + Af = Bu. Assume that 𝛱 is

a projection and that you can control both the systems

 𝛛t f + 𝛱Af = 𝛱Bu,

 𝛛t f + (1-𝛱)Af = (1-𝛱)Bu.

Can you conclude that the first system itself is controllable ? We

cannot expect it in general. But in a joint work with Andreas Hartmann,

we managed to do it for the half-heat equation. It turns out that the

property we need for our case is:

 If 𝛺 satisfies some cone condition, the set {f+g, f∈L²(𝛺), g∈L²(𝛺),

f is holomorphic, g is anti-holomorphic} is closed in L²(𝛺).

The first proof by Friedrichs consists of long computations, and is

very "complex analysis". But a later proof by Shapiro uses quite

general coercivity estimates proved by Smith, whose proof uses some

tools from algebra : Hilbert's nullstellensatz and/or primary ideal

decomposition.

In this first talk, we will introduce the algebraic tools needed and

present Smith's coercivity inequalities. In a second talk, we will

explain how useful these inequalities are to study the control

properties of the half-heat equation.

In this talk, I will present some introductory facts on Hardy-Toeplitz and Bergman-Toeplitz operators. I will also discuss the presence (or absence) of discrete spectrum for a Bergman-Toeplitz operator; this part of the talk will be based on works of Zhao- Zheng et al., 2010- 2020.

Nous présentons plusieurs exemples de fonctions différentiables ayant des propriétés pathologiques. Nous démontrerons en particulier le résultat suivant, obtenu en collaboration avec A. Daniilidis et S. Tapia. Pour tout N≥1, il existe une fonction f de R ^N dans R, localement Lipschitzienne et différentiable en tout point, telle que pour tout compact connexe d'intérieur non vide, il existe x dans R^N tel que K={ lim Df(x_n); (x_n) converge vers x}.

We investigate the connection between the propagation of smallness in two dimensions and one-dimensional spectral estimates. The phenomenon of smallness propagation in the plane, originally obtained by Yuzhe Zhu, reveals how the value of solutions in a small region extends to a larger domain. By revisiting Zhu’s proof, we obtain a quantitative version that includes an explicit dependence on key parameters. This refinement enables us to establish spectral inequalities for one-dimensional Schrödinger operators.

We provide a characterization of universal and multiplier interpolating sequences for de Branges-Rovnyak spaces where the defining function is a non-extreme, rational function. Previous work on this topic examined interpolation in de Branges-Rovnyak spaces specifically in cases where the space coincides with a local Dirichlet space under norm equivalence. The more general setting we are interested in here corresponds to higher order local Dirichlet spaces recently investigated by Gu, Luo and Richter. In this general setting we characterize universal interpolating sequences and show that they coincide with multiplier interpolating sequences. We also explore random interpolation through the use of Steinhaus sequences.

This is work in progress with A. Hartmann.

La problématique générale des espaces atteignables peut être résumée de la manière suivante pour un système contrôlé donné: étant donné un état initial $u_i$ et un temps $T \gt 0$, décrire l'espace $R(u_i,T)$ des états finaux $u_f$ que l'on peut atteindre à partir de $u_i$ au temps $T$. Déterminer l'espace atteignable des systèmes contrôlés est l'un des principaux problèmes de la théorie du contrôle. Donner une caractérisation précise des états qui peuvent être atteints en un certain temps fixé est une question encore largement ouverte pour les systèmes paraboliques: même pour l'équation de la chaleur à coefficients constants en une dimension et contrôlée depuis la frontière, la caractérisation complète de l'espace atteignable, en termes d'espaces de Bergman, n'a été obtenue que très récemment. Basé sur un travail en commun avec Sylvain Ervedoza, je présenterai des résultats sur l’espace atteignable pour l’équation de la chaleur avec des perturbations d’ordre inférieur ou des semi-linéaire en dimension $d\geq 1$.

After explaining the notions of symmetry and differential equations, we review possibilities of symmetry methods and advantages of their usage in the theory of differential equations and mathematical physics.

As a specific example, we discuss the history of the (real potential symmetric) dispersionless  Nizhnik equation and its applications and overview its extended symmetry analysis carried out in our papers. More specifically, we construct essential megaideals of the maximal Lie invariance algebra of this equation. Using the original version of the algebraic megaideals-based method, we compute the point- and contact-symmetry pseudogroups of this equation as well as the point-symmetry pseudogroups of its Lax representation and the original real symmetric dispersionless Nizhnik system. This is the first example in the literature, where there is no need to use the direct method for completing the computation.

In addition, we also find geometric properties of the dispersionless Nizhnik equation that completely define it. Lie reductions of this equation are classified, which results in wide families of its new closed-form invariant solutions. We also study hidden generalized symmetries, hidden cosymmetries and hidden conservation laws of this equation.

In this talk, I will first present recent results on global Lp Carleman estimates for the Laplacian, as well as their application to the quantification of unique continuation for solutions of the equation Δu=Vu+W1​⋅∇u+div(W2​u), in terms of the norms of the potentials V∈Lq0​, W1​∈Lq1​, and W2​∈Lq2​. This is a joint work with my PhD advisors, Belhassen Dehman and Sylvain Ervedoza.

In the second part of the talk, I will present results on the quantification of unique continuation for solutions of the same equation Δu=Vu+W1​⋅∇u+div(W2​u). Using Thomas Wolff's lemma on Euclidean measures and a refined version of Carleman estimates, we obtain quantification results for the unique continuation of solutions u in terms of the norms of the potentials, including first-order potentials that are more singular in the limit integrability class. In particular, we consider the case where W1​∈Lq1​ and W2​∈Lq2​, with q1​>d and q2​>d.

This is a joint work with Pedro Caro and Sylvain Ervedoza.

The restriction conjecture is one of the cornerstones of modern

harmonic analysis. We will here focus on the

restriction conjecture for the sphere $\mathbb{S}^{d-1}$ of

$\mathbb{R}^{d}$ ($d\geq2$) where the conjecture states that

the Fourier transform extends continuously from $L^p(\mathbb{R}^d)$ to

$L^q(\mathbb{S}^{d-1})$

if and only if $1\leq p\leq\frac{2d}{d+1}$ (< 2) and

$\frac{d+1}{p'}\leq\frac{d-1}{q}$ ($\frac{1}{p}+\frac{1}{p'}=1$).

In particular, if $f\in L^p(\mathbb{R}^d)$ for those $p$'s, one may

restrict its Fourier transform to the sphere, despite the fact that

this set

has measure $0$ (the same is not possible if one replaces the sphere

by the boundary of a cube).

The aim of these 2 talks is

-- to explain the meaning of the conjecture (where do those strange

limitations come from)

-- prove it in dimension 2 (Feffermann and Zygmund in the early 70s)

-- prove it when $q=2$ (Thomas-Stein theorem).

-- give a recent application to Nazarov's uncertainty principle (joint

work with A. Iosevich and A. Mayeli)

The restriction conjecture is one of the cornerstones of modern

harmonic analysis. We will here focus on the

restriction conjecture for the sphere $\mathbb{S}^{d-1}$ of

$\mathbb{R}^{d}$ ($d\geq2$) where the conjecture states that

the Fourier transform extends continuously from $L^p(\mathbb{R}^d)$ to

$L^q(\mathbb{S}^{d-1})$

if and only if $1\leq p\leq\frac{2d}{d+1}$ (< 2) and

$\frac{d+1}{p'}\leq\frac{d-1}{q}$ ($\frac{1}{p}+\frac{1}{p'}=1$).

In particular, if $f\in L^p(\mathbb{R}^d)$ for those $p$'s, one may

restrict its Fourier transform to the sphere, despite the fact that

this set

has measure $0$ (the same is not possible if one replaces the sphere

by the boundary of a cube).

The aim of these 2 talks is

-- to explain the meaning of the conjecture (where do those strange

limitations come from)

-- prove it in dimension 2 (Feffermann and Zygmund in the early 70s)

-- prove it when $q=2$ (Thomas-Stein theorem).

-- give a recent application to Nazarov's uncertainty principle (joint

work with A. Iosevich and A. Mayeli)

Partie 1

Partie 2

We study a coupled PDE-ODE system modeling the small oscillations of a floating cylinder interacting with small water waves. The governing equations are formulated as an abstract wave-type equation on a suitable Hilbert space, and we establish the well-posedness of the associated initial value problem. A key element of the proof is the analysis of a partial Dirichlet-to-Neumann map on an unbounded domain with a non-smooth boundary.

Three ball inequalities are a useful tool in the study of unique continuation properties in the continuum. Our goal is to extend these inequalities to certain discrete lattices, known as periodic graphs. Periodic graphs are graphs in $R^d$ that remain invariant under translations of particular vectors. We prove that such inequalities holds for Schrödinger operators on a family of periodic graph, and for Laplace operators on a wider family.

Les inégalités à trois boules sont un outil utile dans l’étude des propriétés de continuation unique dans le continuum. Notre but est d’étendre ces inégalités à certains treillis discrets, connus sous le nom de graphes périodiques. Les graphes périodiques sont des graphes dans $R^d$ qui restent invariants sous les translations de vecteurs particuliers. Nous prouvons que ces inégalités sont valables pour les opérateurs de Schrödinger sur une famille de graphes périodiques, et pour les opérateurs de Laplace sur une famille plus large.

Consider a reproducing kernel Hilbert space $H$ of functions defined on a domain $\Omega$, with reproducing kernel $K$. Given a sequence $\Lambda$ on $\Omega$, it is possible to associate to it a measure $\mu_\Lambda$. Understanding when this measure is a Carleson measure for $H$ is equivalent to know when the family of normalized reproducing kernels evaluated in the points of the sequence $\Lambda$ forms a Bessel system for $H$.

Our work focuses on understanding when the measure $\mu_\Lambda$ is a Carleson measure when $H$ is the Hardy space in the polydisc. While such measures are well understood and characterized by the ’one-box condition’ for the Hardy space in one complex variable, in the polydisc the situation is more complex, and this condition does not directly generalize.

In this scenario is then useful to use a different approach. We consider random sequences with prescribed radii and we provide a 0-1 law that describes when the measure $\mu_\Lambda$ is a Carleson measure almost surely. Using tools from random matrix theory, specifically a matrix Chernoff's inequality, we actually give the 0-1 law ensuring that the Gramian associated to the sequence $\Lambda$ is almost surely bounded.

This is a joint work with N. Chalmoukis and A. Dayan.

Consider a reproducing kernel Hilbert space $H$ of functions defined on a domain $\Omega$, with reproducing kernel $K$. Given a sequence $\Lambda$ on $\Omega$, it is possible to associate to it a measure $\mu_\Lambda$. Understanding when this measure is a Carleson measure for $H$ is equivalent to know when the family of normalized reproducing kernels evaluated in the points of the sequence $\Lambda$ forms a Bessel system for $H$.

Our work focuses on understanding when the measure $\mu_\Lambda$ is a Carleson measure when $H$ is the Hardy space in the polydisc. While such measures are well understood and characterized by the ’one-box condition’ for the Hardy space in one complex variable, in the polydisc the situation is more complex, and this condition does not directly generalize.

In this scenario is then useful to use a different approach. We consider random sequences with prescribed radii and we provide a 0-1 law that describes when the measure $\mu_\Lambda$ is a Carleson measure almost surely. Using tools from random matrix theory, specifically a matrix Chernoff's inequality, we actually give the 0-1 law ensuring that the Gramian associated to the sequence $\Lambda$ is almost surely bounded.

This is a joint work with N. Chalmoukis and A. Dayan.

TBA

TBA