Poincare Inequality Research Articles

Discrete convolutions of $$\mathrm {BV}$$ functions in quasiopen sets in metric spaces

We study fine potential theory and in particular partitions of unity in quasiopen sets in the case $$p=1$$. Using these, we develop an analog of the discrete convolution technique in quasiopen (instead of open) sets. We apply this technique to show that every function of bounded variation ($$\mathrm {BV}$$ function) can be approximated in the $$\mathrm {BV}$$ and $$L^{\infty }$$ norms by $$\mathrm {BV}$$ functions whose jump sets are of finite Hausdorff measure. Our results seem to be new even in Euclidean spaces but we work in a more general complete metric space that is equipped with a doubling measure and supports a Poincare inequality.

A new fractional order Poincare's inequality with weights

We derive a new Sawyer's type sufficient condition for the fractional order Poincare inequality with weights (integral(Omega) vertical bar f(x) - (f) over bar (v,Omega)vertical bar(q) upsilon(x)dx) ...

The Poincaré inequality and quadratic transportation-variance inequalities

It is known that the Poincare inequality is equivalent to the quadratic transportation-variance inequality (namely $W_{2}^{2}(f\mu ,\mu ) \leqslant C_{V} \mathrm{Var} _{\mu }(f)$), see Jourdain [10] and most recently Ledoux [12]. We give two alternative proofs to this fact. In particular, we achieve a smaller $C_{V}$ than before, which equals the double of Poincare constant. Applying the same argument leads to more characterizations of the Poincare inequality. Our method also yields a by-product as the equivalence between the logarithmic Sobolev inequality and strict contraction of heat flow in Wasserstein space provided that the Bakry-Emery curvature has a lower bound (here the control constants may depend on the curvature bound). Next, we present a comparison inequality between $W_{2}^{2}(f\mu ,\mu )$ and its centralization $W_{2}^{2}(f_{c}\mu ,\mu )$ for $f_{c} = \frac{|\sqrt {f} - \mu (\sqrt {f})|^{2}} {\mathrm{Var} _{\mu }(\sqrt{f} )}$, which may be viewed as some special counterpart of the Rothaus’ lemma for relative entropy. Then it yields some new bound of $W_{2}^{2}(f\mu ,\mu )$ associated to the variance of $\sqrt{f} $ rather than $f$. As a by-product, we have another proof to derive the quadratic transportation-information inequality from Lyapunov condition, avoiding the Bobkov-Gotze’s characterization of the Talagrand’s inequality.

Remarks on a limiting case of Hardy type inequalities

The classical Hardy inequality holds in Sobolev spaces $W_0^{1,p}$ when $1\le p< N$. In the limiting case where $p=N$, it is known that by adding a logarithmic function to the Hardy potential, some inequality which is called the critical Hardy inequality holds in $W_0^{1,N}$. In this note, in order to give an explanation of appearance of the logarithmic function at the potential, we derive the logarithmic function from the classical Hardy inequality with the best constant via some limiting procedure as $p \nearrow N$. And we show that our limiting procedure is also available for the classical Rellich inequality in second order Sobolev spaces $W_0^{2,p}$ with $p \in (1, \frac{N}{2})$ and the Poincare inequality.

Analytic-geometric methods for finite Markov chains with applications to quasi-stationarity

For a relatively large class of well-behaved absorbing (or killed) finite Markov chains, we give detailed quantitative estimates regarding the behavior of the chain before it is absorbed (or killed). Typical examples are random walks on box-like finite subsets of the square lattice $\mathbb Z^d$ absorbed (or killed) at the boundary. The analysis is based on Poincare, Nash, and Harnack inequalities, moderate growth, and on the notions of John and inner-uniform domains.

Multiscale functional inequalities in probability: Concentration properties

In a companion article we have introduced a notion of multiscale functional inequalities for functions $X(A)$ of an ergodic stationary random field $A$ on the ambient space $\mathbb R^d$. These inequalities are multiscale weighted versions of standard Poincare, covariance, and logarithmic Sobolev inequalities. They hold for all the examples of fields $A$ arising in the modelling of heterogeneous materials in the applied sciences whereas their standard versions are much more restrictive. In this contribution we first investigate the link between multiscale functional inequalities and more standard decorrelation or mixing properties of random fields. Next, we show that multiscale functional inequalities imply fine concentration properties for nonlinear functions $X(A)$. This constitutes the main stochastic ingredient to the quenched large-scale regularity theory for random elliptic operators by the second author, Neukamm, and Otto, and to the corresponding quantitative stochastic homogenization results.

Spatial domain decomposition approach to dynamic compensator design for linear space‐varying parabolic MIMO PDEs

This study addresses the problem of dynamic compensator design for exponential stabilisation of linear space-varying parabolic multiple-input–multiple-output (MIMO) partial differential equations (PDEs) subject to periodic boundary conditions. With the aid of the observer-based feedback control technique, an observer-based dynamic feedback compensator, whose implementation requires only a few actuators and sensors active over partial areas of the spatial domain, is constructed such that the resulting closed-loop coupled PDEs is exponentially stable. The spatial domain is divided into multiple subdomains according to the minimum of the actuators' number and the sensors' one. By Lyapunov direct method and two general variants of Poincare-Wirtinger inequality at each subdomain, sufficient conditions for the existence of such feedback compensator are developed and presented in terms of algebraic linear matrix inequalities (LMIs) in space. Based on the extreme value theorem, LMI-based sufficient and necessary conditions are presented for the feasibility of algebraic LMIs in space. Finally, numerical simulation results are presented to support the proposed design method.

Global existence and long time behavior of the Ellipsoidal-Statistical-Fokker-Planck model for diatomic gases

We are concerned with the global existence and long time behavior of the solutions to the ES-FP model for diatomic gases proposed in [ 22 ]. The global existence of the solutions for this model near the global Maxwellian is established by nonlinear energy method based on the macro-micro decomposition. An algebraic convergence rate in time of the solutions to the equilibrium state is obtained by constructing the compensating function. Since the density distribution function \begin{document}$ F(t, x, v, I) $\end{document} also contains energy variable \begin{document}$ I $\end{document} , we derive more general Poincare inequality including variables \begin{document}$ v, I $\end{document} on \begin{document}$ \mathbb{R}^3\times \mathbb{R}^+ $\end{document} to establish the coercivity estimate of the linearized operator.

Sobolev-Poincaré inequalities for differential forms and currents

In this note we collect some results in R^n about (p,q) Poincare and Sobolev inequalities for differential forms obtained in a joint research with Franchi and Pansu. In particular, we focus to the case p=1. From the geometric point of view, Poincare and Sobolev inequalities for differential forms provide a quantitative formulation of the vanishing of the cohomology. As an application of the results obtained in the case p=1 we obtain Poincare and Sobolev inequalities for Euclidean currents.

A sharp Leibniz rule for $${\mathrm {BV}}$$ functions in metric spaces

We prove a Leibniz rule for $${\mathrm {BV}}$$ functions in a complete metric space that is equipped with a doubling measure and supports a Poincare inequality. Unlike in previous versions of the rule, we do not assume the functions to be locally essentially bounded and the end result does not involve a constant $$C\ge 1$$ , and so our result seems to be essentially the best possible. In order to obtain the rule in such generality, we first study the weak* convergence of the variation measure of $${\mathrm {BV}}$$ functions, with quasi semicontinuous test functions.

Coercivity, hypocoercivity, exponential time decay and simulations for discrete Fokker–Planck equations

In this article, we propose and study several discrete versions of homogeneous and inhomogeneous one-dimensional Fokker-Planck equations. In particular, for these dis-cretizations of velocity and space, we prove the exponential convergence to the equilibrium of the solutions, for time-continuous equations as well as for time-discrete equations. Our method uses new types of discrete Poincare inequalities for a two-direction discretization of the derivative in velocity. For the inhomogeneous problem, we adapt hypocoercive methods to the discrete cases.

Isoperimetric Inequalities for Non-Local Dirichlet Forms

Let $(E,{\mathscr{E}} F,\mu )$ be a σ-finite measure space. For a non-negative symmetric measurable function J(x,y) on E × E, consider the quadratic form $$ \mathscr{E}(f,f):= \frac{1}{2}{\int}_{E\times E} (f(x)-f(y))^{2} J(x,y) \mu(\text{\text{d}} x) \mu(\text{\text{d}} y) $$ in L2(μ). We characterize the relationship between the isoperimetric inequality and the super Poincare inequality associated with ${\mathscr{E}}$ . In particular, sharp Orlicz-Sobolev type and Poincare type isoperimetric inequalities are derived for stable-like Dirichlet forms on $\mathbb {R}^{n}$ , which include the existing fractional isoperimetric inequality as a special example.

English

We design an abstract setting for the approximation in Banach spaces of operators acting in duality. A typical example are the gradient and divergence operators in Lebesgue–Sobolev spaces on a bounded domain. We apply this abstract setting to the numerical approximation of Leray-Lions type problems, which include in particular linear diffusion. The main interest of the abstract setting is to provide a unified convergence analysis that simultaneously covers (i) all usual boundary conditions, (ii) several approximation methods. The considered approximations can be conforming, or not (that is, the approximation functions can belong to the energy space of the problem, or not), and include classical as well as recent numerical schemes. Convergence results and error estimates are given. We finally briefly show how the abstract setting can also be applied to other models, including flows in fractured medium, elasticity equations and diffusion equations on manifolds. A by-product of the analysis is an apparently novel result on the equivalence between general Poincare inequalities and the surjectivity of the divergence operator in appropriate spaces.

Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps

Nous considerons les proprietes d’homogeneisation de l’operateur de Laplace discret avec des conductances aleatoires. Nous demontrons l’homogeneisation de l’equation de Poisson discrete et des plus hauts elements du spectre de l’operateur de Dirichlet dans un domaine limite. Nous supposons les conductances stationnaires, ergodiques et strictement positives a plus proches voisins. Compare aux resultats precedents, nous remplacons l’ellipticite uniforme par des conditions d’integrabilite des moments des conductances. De plus, nous autorisons des sauts de tailles arbitraires. En l’absence de sauts longs, les conditions sur les moments sont optimales pour l’homogeneisation spectrale. Elles correspondent a la condition necessaire du theoreme central limite pour les marches aleatoires en conductances aleatoires. Nous utilisons l’homogeneisation spectrale pour demontrer un principe de grandes deviations gele pour le temps local normalise de la marche aleatoire dans une suite croissante de boites. Nos demonstrations sont basees sur un resultat de compacite pour l’energie de Dirichlet, les inegalites de Poincare, l’iteration de Moser et la convergence a deux echelles.

On the Hydrodynamic Stability of a Lennard-Jones Molecular Fluid

As a central concept in fluid dynamics stability is fundamental in understanding transitions from laminar to turbulent flow. In continuum flows, it is well-established that a transition to turbulence can occur at subcritical Reynolds numbers, in contrast to theoretical predictions. In non-equilibrium molecular dynamics (NEMD), it has been widely observed that at a critical Reynolds number the fluid undergoes an ordering transition from an amorphous phase to a ‘string’ phase. Using the fluctuation theorem (FT) and the dissipation function, we generalize the classical continuum Reynolds-Orr equation to sheared molecular fluids by ascribing a natural description to the nature of stochastic perturbations, i.e. fluctuations in shear stress. Via the Poincare inequality, we arrive at a new stability criterion by providing a lower bound on the exponential decay of perturbations, which reduces to the classical continuum result in the limit of infinite system size. We investigate the nature of these velocity perturbations and conditions necessary for growth in the kinetic energy of perturbations. We obtain a fluid dependent estimate for the critical Reynolds number by which one may estimate the critical Reynolds number at which the fluid transitions to the string phase, thus providing a framework for generalizing classical continuum theories to the microscale.

Static Collocated Piecewise Fuzzy Control Design of Quasi-Linear Parabolic PDE Systems Subject to Periodic Boundary Conditions

This paper presents a Lyapunov and partial differential equation (PDE)-based methodology to solve static collocated piecewise fuzzy control design of quasi-linear parabolic PDE systems subject to periodic boundary conditions. Two types of piecewise control, i.e., globally piecewise control and locally piecewise control are considered, respectively. A Takagi–Sugeno (T–S) fuzzy PDE model that is constructed via local sector nonlinearity method is first employed to accurately describe spatiotemporal dynamics of quasi-linear PDEs. Based on the T–S fuzzy PDE model, a static collocated piecewise fuzzy feedback controller is constructed to guarantee the locally exponential stability of the resulting closed-loop system. Sufficient conditions for the existence of such fuzzy controller are developed by applying vector-valued Poincare–Wirtinger inequality and its variants and a linear matrix inequality (LMI) relaxation technique. These sufficient conditions are presented in terms of standard LMIs. Finally, the performance of the suggested fuzzy controller is illustrated by numerical simulation results of a nonlinear PDE system described by quasi-linear FitzHugh–Nagumo equation with periodic boundary conditions.

The Two Hyperplane Conjecture

We introduce a conjecture that we call the Two Hyperplane Conjecture, saying that an isoperimetric surface that divides a convex body in half by volume is trapped between parallel hy-perplanes. The conjecture is motivated by an approach we propose to the Hots Spots Conjecture of J. Rauch using deformation and hipschitz bounds for level sets of eigenfunctions. We will relate this approach to quantitative connectivity properties of level sets of solutions to elliptic variational problems, including isoperimetric inequalities, Poincare inequalities, Harnack inequalities, and NTA (non-tangentially accessibility). This paper mostly asks questions rather than answering them, while recasting known results in a new light. Its main theme is that the level sets of least energy solutions to scalar variational problems should be as simple as possible.

Improved multipolar Poincaré–Hardy inequalities on Cartan–Hadamard manifolds

We prove a family of improved multipolar Poincare–Hardy inequalities on Cartan–Hadamard manifolds. For suitable configurations of poles, these inequalities yield an improved multipolar Hardy inequality and an improved multipolar Poincare inequality such that the critical unipolar singular mass is reached at any pole.

Existence of Stein kernels under a spectral gap, and discrepancy bounds

Nous prouvons l’existence de noyaux de Stein pour les mesures de probabilites sur $\mathbb{R}^{d}$ satisfaisant une inegalite de Poincare, et obtenons des bornes sur la discrepance de Stein de telles mesures. Des applications au theoreme central limite sont donnees, dont une nouvelle borne sur la vitesse de convergence en distance de Kantorovitch–Wasserstein $W_{2}$ avec un taux et une dependance en la dimension optimales. Comme corollaire, nous obtenons une version quantitative d’une borne sur la constante de Poincare de mesures de probabilites satisfaisant une contrainte sur le moment d’ordre 2. Les resultats sont plus generalement valides dans le cadre de mesures verifiant une inegalite de Poincare a poids inversee. La preuve est basee sur des arguments simples d’analyse fonctionnelle. De plus, nous demontrons deux proprietes generales sur la discrepance de Stein, valide des lors qu’un noyau de Stein existe : la discrepance de Stein est strictement decroissante le long du TCL, et elle controle le moment d’ordre 3 d’un vecteur aleatoire.

Approximate controllability of the wave equation with mixed boundary conditions

We consider initial boundary-value problem for acoustic equation in the time space cylinder Ω × (0; 2T) with unknown variable speed of sound, zero initial data, and mixed boundary conditions. We assume that (Neumann) controls are located at some part Σ Ω [0; T]; Σ ⊂ 𝜕Ω of the lateral surface of the cylinder Ω × (0; T). The domain of observation is Σ × [0; 2T]; and the pressure on another part (𝜕ΩnΣ) × [0; 2T]) is assumed to be zero for any control. We prove the approximate boundary controllability for functions from the subspace V ⊂ H1(Ω) whose traces have vanished on Σ provided that the observation time is 2T more than two acoustic radii of the domain Ω. We give an explicit procedure for solving Boundary Control Problem (BCP) for smooth harmonic functions from V (i.e., we are looking for a boundary control f which generates a wave uf such that uf (., T) approximates any prescribed harmonic function from V ). Moreover, using the Friedrichs–Poincare inequality, we obtain a conditional estimate for this BCP. Note that, for solving BCP for these harmonic functions, we do not need the knowledge of the speed of sound.