Poincare inequality.

From Poincar\'e Inequalities to Nonlinear Matrix Concentration. June 2020. This paper deduces exponential matrix concentration from a Poincar\'e inequality via a short, conceptual argument. Among ...Poincaré--Friedrichs inequalities for piecewise H1 functions are established. They can be applied to classical nonconforming finite element methods, mortar methods, and discontinuous Galerkin methods.These are quite different things. On one hand, an hourglass-shaped surface, without top and bottom lids, admits a nice zero-boundary inequality, but a lousy zero-mean inequality (I'm measuring niceness by size of constant). The latter is because a function can be 1 1 in the bottom half and −1 − 1 in the upper half, with transition in the ...A modified version of Poincare inequality. 2. Counter example for Poincare inequality does not hold on unbounded domain. 5. Poincare-like inequality. 1. Poincare (Wirtinger) Inequality vanishing on subset of boundary? 8 "Moral" difference between Poincare and Sobolev inequalities. 1.Bernoulli 25(1), 2019, 341-374 https://doi.org/10.3150/17-BEJ989 On the convex Poincaré inequality and weak transportation inequalities RADOSŁAW ADAMCZAK* and ...

数学中，庞加莱不等式（英語： Poincaré inequality ）是索伯列夫空间理论中的一个结果，由法国 数学家 昂利·庞加莱命名。这个不等式说明了一个函数的行为可以用这个函数的变化率的行为和它的定义域的几何性质来控制。也就是说，已知函数的变化率和定义域 ...Poincaré--Friedrichs inequalities for piecewise H1 functions are established. They can be applied to classical nonconforming finite element methods, mortar methods, and discontinuous Galerkin methods.数学中，庞加莱不等式（英語： Poincaré inequality ）是索伯列夫空间理论中的一个结果，由法国 数学家 昂利·庞加莱命名。 这个不等式说明了一个函数的行为可以用这个函数的变化率的行为和它的定义域的几何性质来控制。 也就是说，已知函数的变化率和定义域的情况下，可以对函数的上界作出估计。

$\begingroup$ @BenMcKay Admittedly that's a liberal interpretation of the question, but I took to mean 'Which manifolds admit a Poincare inequality (with $\lambda_1 > 0$)?' I admit I don't know much about this, but I think the question is not so simple in the non-compact case, for complete manifolds say.

The inequality provides the sharp upper bound on convex domains, in terms of the diameter alone, of the best constants in Poincaré inequality. The key point is the implementation of a refinement of the classical Pólya-Szegö inequality for the symmetric decreasing rearrangement which yields an optimal weighted Wirtinger inequality.his Poincare inequality discussed previously [private communication]. The conclusion of Theorem 4 is analogous to the conclusion of the John-Nirenberg theorem for functions of bounded mean oscillation. I would like to thank Gerhard Huisken, Neil Trudinger, Bill Ziemer, and particularly Leon Simon, for helpful comments and discussions. NOTATION.Given a bounded open subset Ω of R n, we establish the weak closure of the affine ball B p A (Ω) = {f ∈ W 0 1, p (Ω): E p f ≤ 1} with respect to the affine functional E p f introduced by Lutwak, Yang and Zhang in [46] as well as its compactness in L p (Ω) for any p ≥ 1.These points use strongly the celebrated Blaschke-Santaló inequality. As counterpart, we develop the basic theory ...The latter inequality allows us to recover by different techniques some weighted Poincaré inequalities previously established in Bobkov and Ledoux [12] for the Beta distribution or in Bonnefont, Joulin and Ma [14] for the generalized Cauchy distribution and to highlight new ones, considering for instance Pearson's class of distributions.

Hence the best constant of Poincare inequality is just $1/\lambda_1$? Am I correct? I think this problem has been well studied. So if you know where I can find a good reference, please kindly direct me there. Thank you! sobolev-spaces; calculus-of-variations; Share. Cite. Follow

The weighted Poincaré inequalities in weighted Sobolev spaces are discussed, and the necessary and sufficient conditions for them to hold are given. That is, the Poincaré inequalities hold if, and only if, the ball measure of non-compactness of the natural embedding of weighted Sobolev spaces is less than 1. ... The weighted Poincare ...

Use Hoelder inequality. Share. Cite. Follow answered Dec 14, 2021 at 10:51. Son Gohan Son Gohan. 4,277 2 2 gold badges 5 5 silver badges 23 23 bronze badges $\endgroup$ 5 $\begingroup$ Can you elaborate some more? How would I use Hoelder? $\endgroup$ - Silver54.MATHEMATICS OF COMPUTATION Volume 80, Number 273, January 2011, Pages 119–140 S 0025-5718(2010)02296-3 Article electronically published on July 8, 2010Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack ExchangeThe sharp Sobolev type inequalities in the Lorentz–Sobolev spaces in the hyperbolic spaces. Journal of Mathematical Analysis and Applications, Vol. 490, Issue. 1, p. 124197. Journal of Mathematical Analysis and Applications, Vol. 490, Issue. 1, p. 124197.May 8, 2002 · The case q = np/(n−p) requires the Sobolev inequality explic-itly for the proof, and thus the inequality can be called the Poincar´e-Sobolev inequality in this case. The domain Ω is required to have the “cone property” (see, e.g., [2]); i.e., each point of Ω is the vertex of a spherical cone with fixed height and angle, which is ...

The Poincaré inequality need not hold in this case. The region where the function is near zero might be too small to force the integral of the gradient to be large enough to control the integral of the function. For an explicit counterexample, let. Ω = {(x, y) ∈ R2: 0 < x < 1, 0 < y < x2} Ω = { ( x, y) ∈ R 2: 0 < x < 1, 0 < y < x 2 } In this paper we mainly prove weighted Poincare inequalities for vector fields satisfying Hormander's condition. A crucial part here is that we are able to get a pointwise estimate for any function over any metric ball controlled by a fractional integral of certain maximal function. The Sobolev type inequalities are also derived. As applications of these weighted inequalities, we will show the ...WEIGHTED POINCARÉ INEQUALITY AND RIGIDITY OF COMPLETE MANIFOLDS BY PETER LI 1 AND JIAPING WANG 2 ABSTRACT. - We prove structure theorems for complete manifolds satisfying both the Ricci curvature lower bound and the weighted Poincaré inequality. In the process, a sharp decay estimate for the minimal positive Green's function is obtained.As it happens, the trace Poincaré inequality is equivalent to an ordinary Poincaré inequality. We are grateful to Ramon Van Handel for this observation. The same result has recently appeared in the independent work of Garg et al. [GKS20]. Proposition 2.4 (Equivalence of Poincaré inequalities). Consider a Markov process (/B: B ≥ 0) ⊂ ΩWe establish functional inequalities on the path space of the stochastic flow x ↦ X t x including gradient inequalities, log-Sobolev inequalities and Poincaré inequalities. These inequalities are shown to be equivalent to bounds on the horizontal Ricci operator Ric H: H → H which is defined taking the trace of the curvature tensor only over H.

1. (1) This inequality requires f f to be differentiable everywhere. (2) With that condition, the answer is the linear functions. The challenge is to prove that. (3) You might as well assume n = 1: n = 1: larger values of n n are trivial generalizations because both sides split into sums over the coordinates.

Abstract. The classic Poincaré inequality bounds the Lq L q -norm of a function f f in a bounded domain Ω ⊂Rn Ω ⊂ R n in terms of some Lp L p -norm of its gradient in Ω Ω. We generalize this in two ways: In the first generalization we remove a set Γ Γ from Ω Ω and concentrate our attention on Λ = Ω ∖ Γ Λ = Ω ∖ Γ.MATRIX POINCARE INEQUALITIES AND CONCENTRATION 3´ its scalar counterpart, establishing a matrix concentration inequality is reduced to proving a matrix Poincar´e inequality. To this aim, for a given probability measure, the main task lies in designing the appropriate Markov generator and calculating the corresponding matrix carr´e du champ ...Poincar´e inequality, this paper studies the weaker Orlicz–Poincar´e inequality. More precisely, for any Young function Φ whose growth is slower than quadric, the Orlicz–Poincar´e inequality f 2 Φ CE(f,f),µ(f):= f dµ =0 is studied by using the well-developed weak Poincar´e inequalities, where E is a conservative DirichletPoincare type inequality along the boundary. 0. Poincare inequality together with Cauchy-Schwarz. Hot Network Questions For large commercial jets is it possible to land and slow sufficiently to leave the runway without using reverse thrust or …1 Answer. for some constant α α. If the bilinear form has a term similar to the left side of your inequality, then using by using the inequality we would be making it smaller by getting to the H1 H 1 norm, which is the opposite of our goal. If the bilinear form has a term similar to the right side of your inequality, most often we could ...This paper is devoted to investigate an interpolation inequality between the Brezis-Vázquez and Poincaré inequalities (shortly, BPV inequality) on nonnegatively curved spaces. As a model case, we first prove that the BPV inequality holds on any Minkowski space, by fully characterizing the existence and shape of its extremals. ...We prove that complete Riemannian manifolds with polynomial growth and Ricci curvature bounded from below, admit uniform Poincaré inequalities. A global, uniform Poincaré inequality for horospheres in the universal cover of a closed, n -dimensional Riemannian manifold with pinched negative sectional curvature follows as a corollary. Comments:A NOTE ON WEIGHTED IMPROVED POINCARÉ-TYPE INEQUALITIES 2 where C > 0 is a constant independent of the cubes we consider and w is in the class A∞ of all Muckenhoupt weights. The authors remark that, although the A∞ condition is assumed, the A∞ constant, which is defined by (1.3) [w]A∞:= sup Q∈Q

Poincar´e inequality, this paper studies the weaker Orlicz–Poincar´e inequality. More precisely, for any Young function Φ whose growth is slower than quadric, the Orlicz–Poincar´e inequality f 2 Φ CE(f,f),µ(f):= f dµ =0 is studied by using the well-developed weak Poincar´e inequalities, where E is a conservative Dirichlet

The relationship between Lyapunov conditions and functional inequalities of Poincaré type (or- dinary or weak Poincaré introduced in [24]) is studied in details in the recent work [3]. The present paper is thus a complement of [3] for the study of stronger inequalities than Poincaré inequality. Let us also mention the paper [2] which is a ...

It is known that this inequality is valid for bounded John domains if w ∈ Ap (see [DD]). As we will see, this result can be extended for more general weights. For example, for a class of weights introduced in [FKS] where the authors consider the classic Poincaré inequality in weighted norms, (1.6) kϕ−ϕΩ,wkLp w(Ω) ≤ Ck∇ϕkLp w(Ω)In mathematics, inequalities are a set of five symbols used to demonstrate instances where one value is not the same as another value. The five symbols are described as “not equal to,” “greater than,” “greater than or equal to,” “less than”...Weighted fractional Poincaré inequalities via isoperimetric inequalities. Our main result is a weighted fractional Poincaré-Sobolev inequality improving the celebrated estimate by Bourgain-Brezis-Mironescu. This also yields an improvement of the classical Meyers-Ziemer theorem in several ways. The proof is based on a fractional isoperimetric ...This work studies mixtures of probability measures on $\\mathbb{R}^n$ and gives bounds on the Poincaré and the log-Sobolev constant of two-component mixtures provided that each component satisfies the functional inequality, and both components are close in the $χ^2$-distance. The estimation of those constants for a mixture can be far more subtle than it is for its parts. Even mixing Gaussian ...1 Answer. Poincaré inequality is true if Ω Ω is bounded in a direction or of finite measure in a direction. ∥φn∥2 L2 =∫+∞ 0 φ( t n)2 dt = n∫+∞ 0 φ(s)2ds ≥ n ‖ φ n ‖ L 2 2 = ∫ 0 + ∞ φ ( t n) 2 d t = n ∫ 0 + ∞ φ ( s) 2 d s ≥ n. ∥φ′n∥2 L2 = 1 n2 ∫+∞ 0 φ′( t n)2 dt = 1 n ∫+∞ 0 φ′(s)2ds ...This paper deduces exponential matrix concentration from a Poincare inequality via a short, conceptual argument. Among other examples, this theory applies to matrix-valued functions of a uniformly log-concave random vector. The proof relies on the subadditivity of Poincare inequalities and a chain rule inequality for the trace of the matrix Dirichlet form. It also uses a symmetrization ...inequality. This gives rise to what is called a local Poincaré-Sobolev inequality, namely, a Poincaré type inequality for which the power in the integral at the left hand side is larger than the power of the integral at the right hand side. The self-improvement on the regularity of functions is not anThe symmetric exponential measure on ℝ, i.e., the measure with density 1 2 e − | t |, satisfies Poincaré inequality with constant 4. Consequently, the same is true for the measure on ℝ n which is the n-fold product of this measure. The canonical Gaussian measure on ℝ and thus on ℝ n satisfies logarithmic Sobolev inequality with ... AN OPTIMAL POINCARE INEQUALITY IN L1 FOR CONVEX DOMAINS GABRIEL ACOSTA AND RICARDO G. DURAN (Communicated by Andreas Seeger) Abstract. For convex domains ˆRnwith diameter dwe prove kuk L1(!) d 2 kruk L1(!) for any uwith zero mean value on!. We also show that the constant 1=2in this inequality is optimal. 1. Introduction Given a bounded domainPoincaré inequalities have been studied extensively in the non-product case: see, for example, [CW, F, FGaW, FGuW, FLW1, GN, HK, J, JS, LI, L2, MS, SW] and the references cited there. For the product case, (1.1) has been studied in weighted situations for the case of the ordinary gradient in [ST] and [Ch].

$\begingroup$ @Jeff: Thank you for your comment. What's in my mind is actually the mixed Dirichlet-Neumann boundary problem: an elliptic equation with zero on one portion of the boundary and zero normal derivative on the rest of the portion.tain the exact constants in the inequalities. Finally, we apply our result to study solutions of ordinary differential equations with given average value. 1. Introduction. If 1 < q, p < <x>, and -oo < a < b < oo, there is a constant TPA depending on a, b such that the one dimensional Poincaré type inequality / (b i rb i \l/ci / rb \1'phis Poincare inequality discussed previously [private communication]. The conclusion of Theorem 4 is analogous to the conclusion of the John-Nirenberg theorem for functions of bounded mean oscillation. I would like to thank Gerhard Huisken, Neil Trudinger, Bill Ziemer, and particularly Leon Simon, for helpful comments and discussions. NOTATION.We prove that complete Riemannian manifolds with polynomial growth and Ricci curvature bounded from below, admit uniform Poincaré inequalities. A global, uniform Poincaré inequality for horospheres in the universal cover of a closed, n -dimensional Riemannian manifold with pinched negative sectional curvature follows as a corollary. Comments:Instagram:https://instagram. stephen j. warewhat time cvs pharmacy close for lunchs 34th sttimeline of special education laws ThisMarkovchainisirreducibleandreversible,thustheoperatorKdefinedby [K˚](x) = X y2X K(x;y)˚(y) isaself-adjointcontractiononL2(ˇ) withrealeigenvalues 1 = 0 > 1 ...Let Omega be a domain in R (N). It is shown that a generalized Poincare inequality holds in cones contained in the Sobolev space W (1,p (.)) (Omega), where p (.) : (Omega) over bar -> [1,infinity ... kansas warhawkmultiplication regrouping I was wondering how can one extend this prove to prove Sobolev-Poincare inequality: $||u-u_\Omega||_{L^{p*}}\... Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.Apr 29, 2023 · In this paper, we prove capacitary versions of the fractional Sobolev–Poincaré inequalities. We characterize localized variant of the boundary fractional Sobolev–Poincaré inequalities through uniform fatness condition of the domain in \ (\mathbb {R}^n\). Existence type results on the fractional Hardy inequality in the supercritical case ... julie albini Regarding this point, a parabolic Poincaré type inequality for u in the framework of Orlicz space, which is a larger class than the L p space, was derived in [12]. In this paper we obtain Sobolev-Poincaré type inequalities for u with weight w = w ( x, t) in the parabolic A p class and G ∈ L w p ( Ω × I, R n) for some p > 1, in Theorem 3 ...In mathematics, the Poincaré inequality is a result in the theory of Sobolev spaces, named after the France mathematician Henri Poincaré. The inequality allows …A Poincaré inequality on Rn and its application to potential fluid flows in space. Lu , Guozhen; Ou, Biao (2004). Thumbnail. View/Download file.