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 ...For generators of Markov semigroups which lack a spectral gap, it is shown how bounds on the density of states near zero lead to a so-called weak Poincaré inequality (WPI), originally introduced by Liggett (Ann Probab 19(3):935-959, 1991). Applications to general classes of constant coefficient pseudodifferential operators are studied. Particular examples are the heat semigroup and the ...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.POINCAR´E-FRIEDRICHS INEQUALITY FOR PIECEWISE H1 FUNCTIONS 123 (V1)Assumethatthesub-domainsD i,1≤i≤m,ineachlevelhavecomparable areas, i.e., |D i|≈thesame(uptomultiplicativeconstants), for 1 ≤ i ≤ m,orThe 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).In 1999, Bobkov [ 10] has shown that any log-concave probability measure satisfies the Poincaré inequality. Here log-concave means that ν ( dx ) = e −V (x)dx where V is a convex function with values in \ (\mathbb R \cup \ {+ \infty \}\). In particular uniform measures on convex bodies are log-concave.Consider the PDE. ∂tu = Lu ∂ t u = L u. where L = Δ + ∇V ⋅ ∇ L = Δ + ∇ V ⋅ ∇ is a self-adjoint operator. I read that if L L has a spectral gap λ > 0 λ > 0 then " [convergence of the initial condition to the stationary distribution us(x) =e−V(x) u s ( x) = e − V ( x)] easily follows by elementary spectral analysis, or by ...Feb 26, 2016 · But the most useful form of the Poincaré inequality is for W1,p/{constants} W 1, p / { c o n s t a n t s }. This inequality measures the connectivity of the domain in a subtle way. For example, joining two squares by a thin rectangle, we get a domain with very large Poincaré constant, because a function can be −1 − 1 in one square, +1 + 1 ... $\begingroup$ It seems to me that the Poincare inequality on bounded domains is strictly weaker than (GN)S. Could you confirm whether the exponents in the (1) Poincare-Wirtinger inequality for oscillations around the mean on bounded domains (2) Poincare inequality for functions on domains bounded in only one direction, are optimal …MATHEMATICS OF COMPUTATION Volume 80, Number 273, January 2011, Pages 119-140 S 0025-5718(2010)02296-3 Article electronically published on July 8, 2010Poincar e inequalities and geometric bounds themodern era : Lichnerowicz’s bound (1958) (M;g)compact Riemannian manifold normalized Riemannian volume elementTheorem 1. The Poincare inequality (0.1) kf fBk Lp (B) C(n; p)krfkLp(B); B Rn; f 2 C1(R n); where B is Euclidean ball, 1 < n and p = np=(n p), implies (0.2) Z jf jBj B Z fBjpdx c(n; p)diam(B)p jrfjpdx; jBj B Rn; f 2 C1(R n); where B is Euclidean ball and 1 < n. Proof. By the interpolation inequality, we get (0.3) kf fBkp kf fBkp kf fBk1 ; We establish the Sobolev inequality and the uniform Neumann-Poincaré inequality on each minimal graph over B_1 (p) by combining Cheeger-Colding theory and the current theory from geometric measure theory, where the constants in the inequalities only depends on n, \kappa, the lower bound of the volume of B_1 (p).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) ⊂ ΩDec 30, 2017 · While studying two seemingly irrelevant subjects, probability theory and partial differential equations (PDEs),I ran into a somewhat surprising overlap:the Poincaré inequality.On one hand, it is not out of the ordinary for analysis based subjects to share inequalities such as Cauchy-Schwarz and Hölder;on the other hand, the two forms ofPoincaré inequality have quite different applications. Connected by Poincaré Inequality. 11 minute read. Published: December 30, 2017 While studying two seemingly irrelevant subjects, probability theory and partial differential equations (PDEs), I ran into a somewhat surprising overlap: the Poincaré inequality.On one hand, it is not out of the ordinary for analysis based subjects to share …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 ...Poincaré Inequality Stephen Keith ABSTRACT. The main result of this paper is an improvement for the differentiable structure presented in Cheeger [2, Theorem 4.38] under the same assumptions of [2] that the given metric measure space admits a Poincaré inequality with a doubling mea sure. To be precise, it is shown in this paper …I am trying to understand the Corollary 9.19 (Poincaré's inequality) in Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis. Suppose that $1 \le p < \infty$ and $\Omega$ is a bounded open set.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.$\begingroup$ It seems to me that the Poincare inequality on bounded domains is strictly weaker than (GN)S. Could you confirm whether the exponents in the (1) Poincare-Wirtinger inequality for oscillations around the mean on bounded domains (2) Poincare inequality for functions on domains bounded in only one direction, are optimal (for smooth domains even?)?We develop Green's function estimate for manifolds satisfying a weighted Poincare inequality together with a compatible lower bound on the Ricci curvature. The estimate is then applied to establish existence and sharp estimates of the solution to the Poisson equation on such manifolds. As an application, a Liouville property for finite energy holomorphic functions is proven on a class of ...So basically, I have proved the Poincare's inequality for p = 1 case. That is, for u ∈ W 1, 1 ( Ω), I have | | u − u ¯ | | L 1 ≤ C | | ∇ u | | L 1. Here u ¯ is the average of u on Ω. Now I need to get the general p case, i.e., for u ∈ W 1, p ( Ω), there is | | u − u ¯ | | L p ≤ C | | ∇ u | | L p. My professor in class ...On a Poincaré inequality with weight. Let Ω Ω be a bounded convex (non-empty) open subset of Rn R n ( Ω Ω can be as smooth as you like). Is it true that there exists a constant C > 0 C > 0 such that the following holds: Assume given a probability measure ω(x)dx ω ( x) d x with ω ∈ Lp(Ω) ω ∈ L p ( Ω). Then, for any function f f in ...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 …Jun 27, 2023 · In mathematics, the Poincaré inequality [1] is a result in the theory of Sobolev spaces, named after the France mathematician Henri Poincaré. The inequality allows one to obtain bounds on a function using bounds on its derivatives and the geometry of its domain of definition. Such bounds are of great importance in the modern, direct methods ... The first part of the Sobolev embedding theorem states that if k > ℓ, p < n and 1 ≤ p < q < ∞ are two real numbers such that. and the embedding is continuous. In the special case of k = 1 and ℓ = 0, Sobolev embedding gives. This special case of the Sobolev embedding is a direct consequence of the Gagliardo-Nirenberg-Sobolev inequality.For other inequalities named after Wirtinger, see Wirtinger's inequality.. In the mathematical field of analysis, the Wirtinger inequality is an important inequality for functions of a single variable, named after Wilhelm Wirtinger.It was used by Adolf Hurwitz in 1901 to give a new proof of the isoperimetric inequality for curves in the plane. A variety …Jan 6, 2021 · Poincaré-Sobolev-type inequalities involving rearrangement-invariant norms on the entire \(\mathbb R^n\) are provided. Namely, inequalities of the type \(\Vert u-P\Vert _{Y(\mathbb R^n)}\le C\Vert abla ^m u\Vert _{X(\mathbb R^n)}\), where X and Y are either rearrangement-invariant spaces over \(\mathbb R^n\) or Orlicz spaces over \(\mathbb R^n\), u is a \(m-\) times weakly differentiable ... 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 ...We consider a domain $$\\varOmega \\subset \\mathbb {R}^d$$ Ω ⊂ R d equipped with a nonnegative weight w and are concerned with the question whether a Poincaré inequality holds on $$\\varOmega $$ Ω , i.e., if there exists a finite constant C independent of f such that It turns out that it is essentially sufficient that on all superlevel sets of w there hold Poincaré inequalities w.r.t ...This paper deduces exponential matrix concentration from a Poincaré 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 Poincaré inequalities and a chain rule inequality for the trace of the matrix In this paper we will establish different weighted Poincaré inequalities with variable exponents on Carnot-Carathéodory spaces or Carnot groups. We will use different techniques to obtain these inequalities. For vector fields satisfying Hörmander's condition in variable non-isotropic Sobolev spaces, we consider a weight in the variable Muckenhoupt class $% A_{p(\\cdot ),p^{\\ast }(\\cdot ...This paper deduces exponential matrix concentration from a Poincaré 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 Poincaré inequalities and a chain rule inequality for the trace of the matrix Sobolev and Poincare inequalities on compact Riemannian manifolds. Let M M be an n n -dimensional compact Riemannian manifold without boundary and B(r) B ( r) a geodesic ball of radius r r. Then for u ∈ W1,p(B(r)) u ∈ W 1, p ( B ( r)), the Poincare and Sobolev-Poincare inequalities are satisfied.An optimal poincaré inequality for convex domains of non-negative curvature ... ~j An Optimal Poincare Inequality 273 Let k denote the expression in braces in the last line. If we sum the above in- equality over j we obtain 21 ~ f 2 dA >(Tz2/d2) ~ f 2 d a - k A ( Q ) ~. ...We consider the question of whether a domain with uniformly thick boundary at all locations and at all scales has a large portion of its boundary visible from the interior; here, "visibility" indicates the existence of John curves connecting the interior point to the points on the "visible boundary". In this paper, we provide an affirmative answer in the setting of a doubling metric measure ...We examine the validity of the Poincaré inequality for degenerate, second-order, elliptic operators H in divergence form on \({L_2(\mathbf{R}^{n}\times \mathbf{R}^{m})}\).We assume the coefficients are real symmetric and \({a_1H_\delta\geq H\geq a_2H_\delta}\) for some \({a_1,a_2>0}\) where H δ is a generalized Grušin operator,Friedrichs's inequality. In mathematics, Friedrichs's inequality is a theorem of functional analysis, due to Kurt Friedrichs. It places a bound on the Lp norm of a function using Lp bounds on the weak derivatives of the function and the geometry of the domain, and can be used to show that certain norms on Sobolev spaces are equivalent.In mathematics, the Poincaré inequality is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincaré. The inequality allows one to obtain bounds on a function using bounds on its derivatives and the geometry of its domain of definition. Such bounds are of great … See moreIn [17], the author and Milman proved a sharp Poincaré inequality for subsets of (essentially non-branching) MCP(K,N) metric measure spaces, whose diame-ter is bounded from above by D. The current paper is a subsequent work of [17]. We will study the general p-poincaré inequality within the class of spaces ver-ifying measure contraction property.In Section 2, taking the dimension to be one, we establish a covariance inequality that is valid for any measure on R and that indeed generalizes the L1-Poincar´e inequality (1.4). Then we will consider in Section 3 extensions of our covariance inequalities that are related to Lp-Poincar´e inequalities, for p ≥Poincare inequality together with Cauchy-Schwarz. Ask Question Asked 1 year, 11 months ago. Modified 1 year, 11 months ago. Viewed 68 times 0 $\begingroup$ Given the advection ...In 1999, Bobkov [ 10] has shown that any log-concave probability measure satisfies the Poincaré inequality. Here log-concave means that ν ( dx ) = e −V (x)dx where V is a convex function with values in \ (\mathbb R \cup \ {+ \infty \}\). In particular uniform measures on convex bodies are log-concave.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.This 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 ...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 prove that, in dimension one, the Poincare inequality is equivalent to a new transport-chi-square inequality linking the square of the quadratic Wasserstein distance with the … Expand. 8. PDF. Save. Analysis and Geometry of Markov Diffusion Operators. D. Bakry, I. Gentil, M. Ledoux.The inequality is indeed a Poincare inequality, but not the classical one for functions that vanish on the boundary. When $\Omega$ is a bounded Lipschitz domain, Poincare's inequality holds for any subspace $$ S:=\{u\in W^{1,2}(\Omega)\ |\ G(u)=0 \} ...Solving the Yamabe Problem by an Iterative Method on a Small Riemannian Domain. S. Rosenberg, Jie Xu. Mathematics. 2021. We introduce an iterative scheme to solve the Yamabe equation −a∆gu+Su = λu p−1 on small domains (Ω, g) ⊂ R equipped with a Riemannian metric g. Thus g admits a conformal change to a constant scalar….Here we show existence of many subsets of Euclidean spaces that, despite having empty interior, still support Poincaré inequalities with respect to the restricted Lebesgue measure. Most importantly, ...norms on both sides of the inequality is quite natural and along the lines of the results for improved Poincaré inequalities involving the gradient found in [7, 8, 14, 22], we believe that the only antecedent of these weighted fractional inequalities is found in [1, Proposition 4.7], where (1.6) is obtained in a star-shaped domain in the case The author first reviews the classical Korn inequality and its proof. Following recent works of S. Kesavan, P. Ciarlet, Jr., and the author, it is shown how the Korn inequality can be recovered by an entirely different proof. This new proof hinges on appropriate weak versions of the classical Poincaré and Saint-Venant lemma. In fine, both proofs essentially depend on a crucial lemma of J. L ...Poincare Inequality on compact Riemannian manifold. Ask Question Asked 1 year, 11 months ago. Modified 1 year, 10 months ago. Viewed 491 times 1 $\begingroup$ I'm ...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.Poincaré-Sobolev-type inequalities indisputably play a prominent role not only in the theory of Sobolev spaces but also in a wide range of applications in analysis of partial differential equations, calculus of variations, mathematical modeling or harmonic analysis (e.g. [5, 20, 44]).These types of inequalities have been exhaustively studied for decades and have been generalized in many ...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.Oct 19, 2022 · 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. 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.. Visit Stack ExchangeRemark 1.10. The inequality (1.6) can be viewed as an implicit form of the weak Poincar e inequality. Note that setting K= 0 (which is excluded in the theorem) leads to the Poincar e inequality. The power of this result is demonstrated in the following corollary, where the celebrated Nash inequality is obtained as a simple consequence.Abstract. In order to describe L2 -convergence rates slower than exponential, the weak Poincaré inequality is introduced. It is shown that the convergence rate of a Markov semigroup and the corresponding weak Poincaré inequality can be determined by each other. Conditions for the weak Poincaré inequality to hold are presented, which …In the case α ∈ [0,1), we follow the approach used in [8] to prove the Sobolev-Poincaré inequality for John domains, modifying it to include the distance to the boundary in our estimates. For g ∈ L 1 (Ω),let E = braceleftbigg x ∈ Ω: integraldisplay Ω g (y) |x − y| n−1+α dy > t bracerightbigg .In the present paper, we deal withthe weighted Poincark inequalitiesin weighted Sobolev spaces W"lP (fl;x0, xfl) and W"tP (Q; w, w), where R is one-dimensional unbounded domain, and give sufficient conditions for the weighted Poincare inequalities to hold. 2.For other inequalities named after Wirtinger, see Wirtinger's inequality. In the mathematical field of analysis, the Wirtinger inequality is an important inequality for functions of a single variable, named after Wilhelm Wirtinger. It was used by Adolf Hurwitz in 1901 to give a new proof of the isoperimetric inequality for curves in the plane.Aug 15, 2022 · 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. For a doubling measure µ, we characterise when µ supports a Poincaré inequality on the bow-tie, in terms of Poincaré inequalities on the separate parts together ...See also: Poincaré Inequality. Share. Cite. Follow edited Apr 13, 2017 at 12:21. Community Bot. 1. answered Jul 11, 2014 at 20:23. user147263 user147263 $\endgroup$ ... Poincare Inequality on compact Riemannian manifold. 0. Integration by parts on compact, non-orientable Riemannian manifold with boundary.First, I consider the condition that $ \Omega $ is convex and prove the inequality. Now I want to deal with the general case by using the extension theorem of Sobolev space. ... Using the Rellich-Kondrachov theorem to prove Poincare inequality for a function vanishing at one point. 0. Poincaré inequality on annular regions. 4. A Poincaré-type ...An Isoperimetric Inequality for the N-dimensional Free Membrane Problem. J. Rational Mech. Anal. 5, 633-636 (1956). MATH MathSciNet Google Scholar Download references. Author information. Authors and Affiliations. Institute for Fluid Dynamics and Applied Mathematics University of Maryland, College Park, Maryland ...Thus 1/λ1 1 / λ 1 is the best constant in the Poincaré inequality since the infimum is achieved by the solution to the Dirichlet problem. Now, the crucial feature of this is that for a ball, namely Ω = B(0, r) Ω = B ( 0, r), we can explicitly compute the eigenfunctions and eigenvalues of the Laplacian by using the classical PDE technique ...In this paper, we study the sharp Poincaré inequality and the Sobolev inequalities in the higher-order Lorentz-Sobolev spaces in the hyperbolic spaces. These results generalize the ones obtained in Nguyen VH (J Math Anal Appl, 490(1):124197, 2020) to the higher-order derivatives and seem to be new in the context of the Lorentz-Sobolev spaces defined in the hyperbolic spaces.Apr 13, 2018 · 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. where the first implication follows from Paolini and Stepanov's work. As explained above, the second implication follows from [15, Theorem B.15] in the Q-regular case, and in full generality from [8, Chapter 4].Section 4 is the core of the paper, containing the proof of the "only if" implication of Theorem 1.3.In short, the idea is to translate the problem of finding currents in \((X,d ...Abstract In order to describe L 2 -convergence rates slower than exponential, the weak Poincare inequality is introduced. It is shown that the convergence rate of a Markov semigroup and the corresponding weak Poincare inequality can be determined by each other. Conditions for the weak Poincare inequality to hold are presented, which are easy to check and which hold in many applications. The ...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.1 The Dirichlet Poincare Inequality Theorem 1.1 If u : Br → R is a C1 function with u = 0 on ∂Br then 2 ≤ C(n)r 2 u| 2 . Br Br We will prove this for the case n = 1. Here the statement becomes r r f2 ≤ kr 2 (f )2 −r −r where f is a C1 function satisfying f(−r) = f(r) = 0. By the Fundamental Theorem of Calculus s f(s) = f (x). −rStudying the heat semigroup, we prove Li-Yau-type estimates for bounded and positive solutions of the heat equation on graphs. These are proved under the assumption of the curvature-dimension inequality CDE′(n,0){\\mathrm{CDE}^{\\prime}(n,0)}, which can be considered as a notion of curvature for graphs. We further show that non-negatively curved graphs (that is, graphs satisfying CDE ...In this paper we unify and improve some of the results of Bourgain, Brezis and Mironescu and the weighted Poincaré-Sobolev estimate by Fabes, Kenig and Serapioni. More precisely, we get weighted counterparts of the Poincaré-Sobolev type inequality and also of the Hardy type inequality in the fractional case under some mild natural restrictions. A main feature of the results we obtain is the ...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 …I tried to prove on my own theorem 2 of chapter 6 of Evans partial differential equations second edition, but my proof, A GENERALIZED POINCARE INEQUALITY FOR GAUSSIAN MEASURES WILLIAM, New inequalities are obtained which interpolate in a sharp way between the Po, Jan 1, 2021 · In different from Sobolev’s inequality, the geometry of domain is essential for P, My thoughts/ideas: I looked at the case that v ( x) = ∫ a x v ˙ ( t) d t. By Schwarz inequality, What kind of Poincare inequality is that, in which the derivative lies on the left hand-side? Is $\partial_X^{-1, Aug 31, 2017 · Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the lar, To set up Poincaré’s inequality constraint, firs, As BaronVT notes, in order to do something in the frequency sp, Poincaré inequality In mathematics, the Poincaré inequali, Poincaré inequality. Download conference paper PDF. 1 In, Poincare Inequality The Sobolev inequality Ilulinp/(n-p) , Theorem 1. ForanysimpleconnectedgraphG,if isasetofcanonicalpathst, Cheeger, Hajlasz, and Koskela showed the importance of local Poincar&#, Poincaré inequality in a ball (case $1\leqslant p < \in, The relationship between Lyapunov conditions and functio, The main contribution is the conditional Poincar{ , The classical proof for the Poincaré inequality. uL2(Ω) ≤ cΩ ∇uL2(Ω).