Poincare inequality

Poincare Inequalities in Punctured Domains. The classic Poincare inequality bounds the Lq -norm of a function f in a bounded domain $\Omega \subset \R^n$ in terms of some Lp -norm of its gradient in Ω. We generalize this in two ways: In the first generalization we remove a set Γ from Ω and concentrate our attention on Λ = Ω ∖ Γ.

We show that any probability measure satisfying a Matrix Poincaré inequality with respect to some reversible Markov generator satisfies an exponential matrix concentration inequality depending on the associated matrix carré du champ operator. This extends to the matrix setting a classical phenomenon in the scalar case. Moreover, the proof gives rise to new matrix trace inequalities which ...Take the square of the inverse of (4a 2 r 2 + 1 e + 2)m (r − 1) as 1 2 β (s) for the desired conclusion. a50 In [24] Eberle showed that a local Poincaré inequality holds for loops spaces over a compact manifold. However the computation was difficult and complicated and there wasn’t an estimate on the blowing up rate.1 Answer. Finding the best constant for Poincare inequality (or korn's inequality) is a long standing problem. Unfortunately, there is no general answer. (not I am known of). However, for some specially domains, there is something you can do. For example, if Ω Ω is a ball, then the best constant is the radius of the ball (or something …A Poincare Inequality on Loop Spaces´ Xin Chen, Xue-Mei Li and Bo Wu Mathemtics Institute University of Warwick Coventry CV4 7AL, U.K. November 9, 2018 Abstract We investigate properties of measures in infinite dimension al spaces in terms of Poincare´ inequalities. A Poincare´ inequality states that the L2 vari-Indeed, such estimates are directly related to well-known inequalities from pure mathematics (e.g logarithmic Sobolev and Poincáre inequalities). In probability theory, Brascamp–Lieb and Poincaré inequalities are two very important concentration inequalities, which give upper bounds on variance of function of random variables.I think that this is known as some version of ``Poincare's inequality''. multivariable-calculus; sobolev-spaces; Share. Cite. Follow asked Apr 11, 2012 at 23:12. Stefan Smith Stefan Smith. 7,882 2 2 gold badges 40 40 silver badges 61 61 bronze badges $\endgroup$ 3Keywords: Ergodic processes; Lyapunov functions; Poincaré inequalities; Hypocoercivity 1. Introduction, framework and first results Rate of convergence to equilibrium is one of the most studied problem in various areas of mathematics and physics. In the present paper we shall consider a dynamics given by a time * Corresponding author at ...

Mar 19, 2021 · In Evans PDE book there is the following theorem: (Poincaré's inequality for a ball). Assume 1 ≤ p ≤ ∞. 1 ≤ p ≤ ∞. Then there exists a constant C, C, depending only on n n and p, p, such that. ∥u − (u)x,r∥Lp(B(x,r)) ≤ Cr∥Du∥Lp(B(x,r)) ‖ u − ( u) x, r ‖ L p ( B ( x, r)) ≤ C r ‖ D u ‖ L p ( B ( x, r)) The ... Weighted Poincaré inequalities for Hörmander vector fields and local regularity for a class of degenerate elliptic equations. B. Franchi G. Lu R. Wheeden. Mathematics. 1995. In this note we state weighted Poincaré inequalities associated with a family of vector fields satisfying Hörmander rank condition.The Poincaré inequality and these Sobolev spaces are then applied to produce various results on existence, uniqueness and qualitative properties of weak solutions to boundary-value problems for degenerate elliptic, degenerate parabolic and degenerate hyperbolic partial differential equations (PDEs) of second order written in divergence form ...A new approach is proposed for proving such inequalities in bounded convex domains. Quite a number of works are available, where a sufficient condition on weight functions is proved for a Poincaré type inequality to hold (see e.g. [ 6, 11, 24, 28 ]). In the present paper, we give a Sawyer type sufficient condition (see e.g. [ 27, 28 ]).Function approximation and recovery via some sampled data have long been studied in a wide array of applied mathematics and statistics fields. Analytic tools, such as the Poincaré inequality, have been handy for estimating the approximation errors in different scales. The purpose of this paper is to study a generalized Poincaré inequality, where the measurement function is of subsampled type ...THE POINCARE INEQUALITY IS AN OPEN ENDED CONDITION´ 577 Corollary 1.0.2. Let p>1 and let w be a p-admissible weight in Rn, n ≥ 1. Then there exists ε>0 such that w is q-admissible for every q>p−ε, quantitatively. For complete Riemannian manifolds, Saloff-Coste ([41], [42]) established

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) ⊂ ΩThis caused me to investigate the 1913 edition of Websters Dictionary - which is now in the public domain. However, after a day's work wrangling it into a ...weak Poincare inequality for geodesic balls. The weak Poincare inequality asserts that C inf f (u -a)2 < f VIVU2, aE B(R) B(2R) for any function u E H1,2(B(2R)) and for some constant C > 0 depending only on R, the dimension m, and the lower bound of the Ricci curvature on B(2R). However, a covering argument in [J] asserts that the weak Poincare ...energy side of the Poincare´ inequality. In particular, our results apply to geodesic balls in ideal sub-Riemannian manifolds, such as the Heisenberg group. Mathematics Subject Classification (2010): 35P15 (primary); 58J50, 53C23, 51F99 (secondary). 1. I ntr oduc i Determining the optimal constant in the Poincar´e inequality, which in an ...For what it's worth, I'm looking at the book and Evans writes "This estimate is sometimes called Poincare's inequality." (Page 282 in the second edition.) See also the Wiki article or Wolfram Mathworld, which have somewhat divergent opinions on what should or shouldn't be called a Poincare inequality.

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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.The Poincaré, or spectral gap, inequality is the simplest inequality which quantifies ergodicity and controls convergence to equilibrium of the semigroup P = ( P t ) t≥0 …A Poincaré inequality states that the variance of an admissible function is controlled by the homogeneous norm. In the case of Loop spaces, it was observed by L. Gross that the homogeneous norm alone may not control the norm and a potential term involving the end value of the Brownian bridge is introduced. Aida, on the other hand, introduced a ...Aug 31, 2017 · 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. examples which show that this inequality is false for all p < 1, even if q is very small, Ω is a ball, and u is smooth (one such example is given near the end of Section 1). Nevertheless, we shall show that, under a rather mild condition on ∇u, one can prove such an inequality in any John domain for all 0 < p < 1 (see Theorem 1.5).

Matteo Levi, Federico Santagati, Anita Tabacco, Maria Vallarino. We prove local Lp -Poincaré inequalities, p ∈ [1, ∞], on quasiconvex sets in infinite graphs endowed with a family of locally doubling measures, and global Lp -Poincaré inequalities on connected sets for flow measures on trees. We also discuss the optimality of our results.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”...Poincare Inequality 635 which completes the proof. Next, we give a criterion for sets which are not P-domains. Definition 2. An open subset Ω in RN is called a B-domain if for any r>0, there exists an open ball of radius r contained in Ω. Theorem 2. Any B-domain is not a P-domain. In the proof of Theorem 2, we will use the following lemma ...Matteo Levi, Federico Santagati, Anita Tabacco, Maria Vallarino. We prove local Lp -Poincaré inequalities, p ∈ [1, ∞], on quasiconvex sets in infinite graphs endowed with a family of locally doubling measures, and global Lp -Poincaré inequalities on connected sets for flow measures on trees. We also discuss the optimality of our results.Abstract. In an \(n\)-dimensional bounded domain \(\Omega_n\), \(n\ge 2\), we prove the Steklov-Poincaré inequality with the best constant in the case where \(\Omega_n\) is an \(n\)-dimensional ball.We also consider the case of an unbounded domain with finite measure, in which the Steklov-Poincaré inequality is proved on the basis of a Sobolev inequality.Lecture Five: The Cacciopolli Inequality The Cacciopolli Inequality The Cacciopolli (or Reverse Poincare) Inequality bounds similar terms to the Poincare inequalities studied last time, but the other way around. The statement is this. Theorem 1.1 Let u : B 2r → R satisfy u u ≥ 0. Then | u| ≤2 4 2 r B 2r \Br u . (1) 2 Br First prove a Lemma.In this set up, can one still conclude Poincare inequality? i.e. does the following hold? $$ \lVert u \rVert_{L^p(D)} < C \lVert \nabla u \rVert_{L^p(D)} \quad \forall u \in W$$ Having reviewed Evan's book amongst others, I did not seem to find a result concerning this case, any suggestion would be most helpful. Can one, perhaps, as in …POINCARE INEQUALITIES 5 of a Sobolev function uis, up to a dimensional constant, the minimal that can be inserted to the Poincar e inequality. This is proved along with the characterization in [15]. All of the previous examples share the common feature of exhibiting a self-improving property. Namely, if the inequalities above hold withA NOTE ON SHARP 1-DIMENSIONAL POINCAR´E INEQUALITIES 2311 Poincar´e inequality to these subdomains with a weight which is a positive power of a nonnegative concave function. Moreover, it has recently been shown in [11] by a similar method that the best constant C in the weighted Poincar´e inequality for 1 ≤ q ≤ p<∞, f − f av Lq w (Ω ...There exists an open set of data satisfying the indicated required conditions, obtained by first choosing $\lambda_0$ greater than some constant linked with the Poincaré inequality of the manifold $(S, \sigma)$." Here, I don't really know how to use this inequality. If I could have some sort of inequality

Every graph of bounded degree endowed with the counting measure satisfies a local version of Lp-Poincaré inequality, p ∈ [1, ∞]. We show that on graphs which are trees the Poincaré constant grows at least exponentially with the radius of balls. On the other hand, we prove that, surprisingly, trees endowed with a flow measure support a …

The inequality (3.3) follows from (3.12) and (3.13) and the theorem is proved. a50 We call inequality (3.3) a “weighted Poincaré-type inequality for stable processes.” It is interesting to note that the eigenfunction ϕ 1 in (3.3) can be replaced by various other simi- larly generated functions from P x {τ D >t}. For example, we may ...Mar 19, 2021 · In Evans PDE book there is the following theorem: (Poincaré's inequality for a ball). Assume 1 ≤ p ≤ ∞. 1 ≤ p ≤ ∞. Then there exists a constant C, C, depending only on n n and p, p, such that. ∥u − (u)x,r∥Lp(B(x,r)) ≤ Cr∥Du∥Lp(B(x,r)) ‖ u − ( u) x, r ‖ L p ( B ( x, r)) ≤ C r ‖ D u ‖ L p ( B ( x, r)) The ... A NOTE ON POINCARE- AND FRIEDRICHS-TYPE INEQUALITIES 5 3. Poincar e-type inequalities in Hm() Now we consider Poincar e-type inequalities in Hm() with m2N 0. Throughout this section let ˆRdbe a bounded domain with Lipschitz boundary. On Hm() we use the inner product (u;v) m= X jsj m Z DsuDsvdx and the induced norm kkA 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∈QOn the other hand, we prove that, surprisingly, trees endowed with a flow measure support a global version of Lp -Poincaré inequality, despite the fact that they …POINCARE INEQUALITIES ON RIEMANNIAN MANIFOLDS 79. AIso if the multiplicity of 11, is Qreater than I , then-12. nt' ' a2. The proofs of Theorems 3 and 4 are based on inequalities for the first.5 - Poincaré inequality and the first eigenvalue. Published online by Cambridge University Press: 05 June 2012. Peter Li.

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Poincaré--Friedrichs inequalities for piecewise H1 functions are established. They can be applied to classical nonconforming finite element methods, ...Sep 15, 2020 · Hardy and Poincaré inequalities in fractional Orlicz-Sobolev spaces. Kaushik Bal, Kaushik Mohanta, Prosenjit Roy, Firoj Sk. We provide sufficient conditions for boundary Hardy inequality to hold in bounded Lipschitz domains, complement of a point (the so-called point Hardy inequality), domain above the graph of a Lipschitz function, the ... We point out some of the differences between the consequences of p-Poincaré inequality and that of ∞-Poincaré inequality in the setting of doubling metric measure spaces. Based on the geometric characterization of ∞-Poincaré inequality given in Durand-Cartagena et al. (Mich Math J 60, 2011), we obtain a geometric property implied by the support of a p-Poincaré inequality, and ...Racial, gender, age and socio-economic inequalities lead to discrimination against some people everyday. These inequalities are present in such aspects as education, the workplace, politics, community and even health care.For example, I believe one can extend u u to an H2 H 2 function with compact support in a ball in R2 R 2 and then use a Poincare inequality in the ball. The extension however is not easy. A more direct proof would use the fundamental theorem of calculus on many segments in the domain, but then there you have to do potentially complicated geometry.The main contribution is the conditional Poincar{\'e} inequality (PI), which is shown to yield filter stability. The proof is based upon a recently discovered duality which is used to transform the nonlinear filtering problem into a stochastic optimal control problem for a backward stochastic differential equation (BSDE). Based on these dual ...Jul 8, 2010 · MATHEMATICS OF COMPUTATION Volume 80, Number 273, January 2011, Pages 119–140 S 0025-5718(2010)02296-3 Article electronically published on July 8, 2010 2 Answers. where fΩ =∫Ω f f Ω = ∫ Ω f is the mean of f f. This is exactly your first inequality, but I think (1) captures the meaning better. The weighted Poincaré inequality would be. where fΩ,w =∫Ω fw f Ω, w = ∫ Ω f w is the weighted mean of f f. Again, this is what you have but written in a more natural way.Lipschitz Domain. Dyadic Cube. Bound Lipschitz Domain. Common Face. Uniform Domain. We show that fractional (p, p)-Poincaré inequalities and even fractional Sobolev-Poincaré inequalities hold for bounded John domains, and especially for bounded Lipschitz domains. We also prove sharp fractional (1,p)-Poincaré inequalities for s-John domains.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.If μ satisfies the inequality SG(C) on Rd then (1.3) can be rewritten in a more pleasant way: for all subset A of (Rd)n with μn(A)≥1/2, ∀h≥0 μn A+ √ hB2 +hB1 ≥1 −e−hL (1.4) with a constant L depending on C and the dimension d. The archetypic example of a measure satisfying the classical Poincaré inequality is the exponential ... ….

Cheeger, Hajlasz, and Koskela showed the importance of local Poincaré inequalities in geometry and analysis on metric spaces with doubling measures in [9, 15].In this paper, we establish a family of global Poincaré inequalities on geodesic spaces equipped with Borel measures, which satisfy a local Poincaré inequality along with certain other geometric conditions.My thoughts/ideas: I looked at the case that v ( x) = ∫ a x v ˙ ( t) d t. By Schwarz inequality I get the following: v ( x) 2 ≤ ( x − a) ‖ v ˙ ‖ L 2 ( Ω) 2. If I integrate both sides and take the square root I get exactly what I wanted to show. However, v ( x) = ∫ a b v ˙ ( t) d t isn't necessarily true.POINCARÉ INEQUALITIES ON RIEMANNIAN MANIFOLDS. BONNESEN-TYPE INEQUALITIES IN ALGEBRAIC GEOMETRY, I: INTRODUCTION TO THE PROBLEM. LIOUVILLE THEOREMS FOR HARMONIC MAPPINGS, AND AN APPROACH TO BERNSTEIN THEOREMS. SUBHARMONIC FUNCTIONS, HARMONIC MAPPINGS AND ISOMETRIC IMMERSIONS. AN ISOPERIMETRIC INEQUALITY AND WIEDERSEHEN MANIFOLDS.A relationship between Poincaré-type inequalities and representation formulas in spaces of homogeneous type, International Math. Research Notices, 1996, 1-14. Franchi B., Wheeden R. L., Some remarks about Poincaré type inequalities and representation formulas in metric spaces of homogeneous type, J. Inequalities and Applications, 1999, 3(1 ...The main contribution is the conditional Poincar{\'e} inequality (PI), which is shown to yield filter stability. The proof is based upon a recently discovered duality which is used to transform the nonlinear filtering problem into a stochastic optimal control problem for a backward stochastic differential equation (BSDE).The Poincare´ inequality. The Poincare´ inequality is said to hold on an open set D ⊂ Rn if there is a constant C >0 such that Z D |f|2 dV ≤ C Xn j=1 Z D ∂f ∂x j 2 dV holds for all f ∈ C∞ c (D), i.e., smooth functions with compact support in D. The best constant in the Poincare´ inequality for an open set D is traditionally denotedHenri Poincaré was a mathematician, theoretical physicist and a philosopher of science famous for discoveries in several fields and referred to as the last polymath, one who could make significant contributions in multiple areas of mathematics and the physical sciences. This survey will focus on Poincaré's philosophy.In functional analysis, the Poincaré inequality says that there exist constants and such that. for all functions in the Sobolev space consisting of all functions in … Poincare inequality, [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1]