Sharp Constant for Poincaré-Type Inequalities in the Hyperbolic Space (original) (raw)

On some strong Poincaré inequalities on Riemannian models and their improvements

Journal of Mathematical Analysis and Applications, 2020

We prove second and fourth order improved Poincaré type inequalities on the hyperbolic space involving Hardy-type remainder terms. Since theirs l.h.s. only involve the radial part of the gradient or of the laplacian, they can be seen as stronger versions of the classical Poincaré inequality. We show that such inequalities hold true on model manifolds as well, under suitable curvature assumptions and sharpness of some constants is also discussed.

On higher order Poincaré inequalities with radial derivatives and Hardy improvements on the hyperbolic space

Annali di Matematica Pura ed Applicata (1923 -), 2021

In this paper we prove higher order Poincaré inequalities involving radial derivatives namely, H N |∇ k r,H N u| 2 dv H N ≥ N − 1 2 2(k−l) H N |∇ l r,H N u| 2 dv H N for all u ∈ H k (H N), where underlying space is N-dimensional hyperbolic space H N , 0 ≤ l < k are integers and the constant N−1 2 2(k−l) is sharp. Furthermore we improve the above inequalities by adding Hardytype remainder terms and the sharpness of some constants is also discussed.

Sharp Adams–Moser–Trudinger type inequalities in the hyperbolic space

Revista Matemática Iberoamericana, 2020

The purpose of this paper is to establish some Adams-Moser-Trudinger inequalities, which are the borderline cases of the Sobolev embedding, in the hyperbolic space H n. First, we prove a sharp Adams' inequality of order two with the exact growth condition in H n. Then we use it to derive a sharp Adams-type inequality and an Adachi-Tanaka-type inequality. We also prove a sharp Adams-type inequality with Navier boundary condition on any bounded domain of H n , which generalizes the result of Tarsi to the setting of hyperbolic spaces. Finally, we establish a Lions-type lemma and an improved Adams-type inequality in the spirit of Lions in H n. Our proofs rely on the symmetrization method extended to hyperbolic spaces.

An optimal improvement for the Hardy inequality on the hyperbolic space and related manifolds

Proceedings, 2019

We prove optimal improvements of the Hardy inequality on the hyperbolic space. Here, optimal means that the resulting operator is critical in the sense of , namely the associated inequality cannot be further improved. Such inequalities arise from more general, optimal ones valid for the operator P λ := -∆ H N -λ where 0 ≤ λ ≤ λ1(H N ) and λ1(H N ) is the bottom of the L 2 spectrum of -∆ H N , a problem that had been studied in only for the operator P λ 1 (H N ) . A different, critical and new inequality on H N , locally of Hardy type, is also shown. Such results have in fact greater generality since they are proved on general Cartan-Hadamard manifolds under curvature assumptions, possibly depending on the point. Existence/nonexistence of extremals for the related Hardy-Poincaré inequalities are also proved using concentration-compactness technique and a Liouville comparison theorem. As applications of our inequalities we obtain an improved Rellich inequality and we derive a quantitative version of Heisenberg-Pauli-Weyl uncertainty principle for the operator P λ . In the case of a Cartan-Hadamard manifold M of dimension N (namely, a manifold which is complete, simply-connected, and has everywhere non-positive sectional curvature), the geodesic distance function d(x, x 0 ), where x 0 ∈ M , satisfies all the assumptions of the weight ̺ and the above inequality holds with best constant N -2 2 2 , see . In particular, considering the most important example of Cartan-Hadamard manifold, namely the hyperbolic space H N , inequality (1.1) reads with r := d(x, x 0 ) and x 0 ∈ H N is a fixed pole.

Extremals for Sobolev and Moser Inequalities in Hyperbolic Space

Milan Journal of Mathematics, 2011

We review some recent results concernig existence/ non existence/ uniqueness of extremals for Sobolev inequalities in Hyperbolic spaces. We also discuss exponential integrability in the hyperbolic plane and related topics.

Some remarks on the Sobolev inequality in Riemannian manifolds

Proceedings of the American Mathematical Society, 2021

We investigate Sobolev and Hardy inequalities, specifically weighted Minerbe’s type estimates, in noncompact complete connected Riemannian manifolds whose geometry is described by an isoperimetric profile. In particular, we assume that the manifold satisfies the p p -hyperbolicity property, stated in terms of a necessary integral Dini condition on the isoperimetric profile. Our method seems to us to combine sharply the knowledge of the isoperimetric profile and the optimal Bliss type Hardy inequality depending on the geometry of the manifold. We recover the well known best Sobolev constant in the Euclidean case.

On the Uniform Poincaré Inequality

Communications in Partial Differential Equations, 2007

We give a proof of the Poincaré inequality in W 1,p (Ω) with a constant that is independent of Ω ∈ U, where U is a set of uniformly bounded and uniformly Lipschitz domains in R n. As a byproduct, we obtain the following : The first non vanishing eigenvalues λ 2 (Ω) of the standard Neumann (variational) boundary value problem on Ω for the Laplace operator are bounded below by a positive constant if the domains Ω vary and remain uniformly bounded and uniformly Lipschitz regular.

A sharp Sobolev inequality on Riemannian manifolds

Comptes Rendus Mathematique, 2002

Let (M,g) be a smooth compact Riemannian manifold without boundary of dimension n>=6. We prove that {align*} \|u\|_{L^{2^*}(M,g)}^2 \le K^2\int_M\{|\nabla_g u|^2+c(n)R_gu^2\}dv_g +A\|u\|_{L^{2n/(n+2)}(M,g)}^2, {align*} for all u\in H^1(M), where 2^*=2n/(n-2), c(n)=(n-2)/[4(n-1)], R_g is the scalar curvature, K−1=inf∣nablau∣L2(Rn)∣u∣L2n/(n−2)(Rn)−1K^{-1}=\inf\|\nabla u\|_{L^2(\R^n)}\|u\|_{L^{2n/(n-2)}(\R^n)}^{-1}K1=infnablau∣_L2(Rn)u∣_L2n/(n2)(Rn)1 and A>0 is a constant depending on (M,g) only. The inequality is {\em sharp} in the sense that on any (M,g), KKK can

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Nonexistence results for parabolic equations involving the {\varvec{p}}$$-Laplacian and Hardy–Leray-type inequalities on Riemannian manifolds

Journal of Evolution Equations, 2021

The main goal of this paper is twofold. The first one is to investigate the nonexistence of positive solutions for the following nonlinear parabolic partial differential equation on a noncompact Riemannian manifold M, ⎧ ⎪ ⎨ ⎪ ⎩ ∂u ∂t = Δ p,g u + V (x)u p−1 + λu q in Ω × (0, T), u(x, 0) = u 0 (x) ≥ 0 i nΩ, u(x, t) = 0 o n ∂Ω × (0, T), where 1 < p < 2, V ∈ L 1 loc (Ω), q > 0, λ ∈ R, Ω is bounded and has a smooth boundary in M and Δ p,g is the p-Laplacian on M. The second one is to obtain Hardy-and Leray-type inequalities with remainder terms on a Riemannian manifold M that provide us concrete potentials to use in the partial differential equation we are interested in. In particular, we obtain explicit (mostly sharp) constants for these inequalities on the hyperbolic space H n .

On some strong Poincaré inequalities on Riemannian models and their improvements

Journal of Mathematical Analysis and Applications, 2020

We prove second and fourth order improved Poincaré type inequalities on the hyperbolic space involving Hardy-type remainder terms. Since theirs l.h.s. only involve the radial part of the gradient or of the laplacian, they can be seen as stronger versions of the classical Poincaré inequality. We show that such inequalities hold true on model manifolds as well, under suitable curvature assumptions and sharpness of some constants is also discussed.

Hardy–Rellich and second order Poincaré identities on the hyperbolic space via Bessel pairs

Calculus of Variations and Partial Differential Equations

We prove a family of Hardy–Rellich and Poincaré identities and inequalities on the hyperbolic space having, as particular cases, improved Hardy-Rellich, Rellich and second order Poincaré inequalities. All remainder terms provided improve those already known in literature, and all identities hold with same constants for radial operators also. Furthermore, as applications of the main results, second order versions of the uncertainty principle on the hyperbolic space are derived.

On higher order Poincaré inequalities with radial derivatives and Hardy improvements on the hyperbolic space

Annali di Matematica Pura ed Applicata (1923 -), 2021

In this paper we prove higher order Poincaré inequalities involving radial derivatives namely, H N |∇ k r,H N u| 2 dv H N ≥ N − 1 2 2(k−l) H N |∇ l r,H N u| 2 dv H N for all u ∈ H k (H N), where underlying space is N-dimensional hyperbolic space H N , 0 ≤ l < k are integers and the constant N−1 2 2(k−l) is sharp. Furthermore we improve the above inequalities by adding Hardytype remainder terms and the sharpness of some constants is also discussed.

Hardy’s Identities and Inequalities on Cartan-Hadamard Manifolds

The Journal of Geometric Analysis

We study the Hardy identities and inequalities on Cartan-Hadamard manifolds using the notion of a Bessel pair. These Hardy identities offer significantly more information on the existence/nonexistence of the extremal functions of the Hardy inequalities. These Hardy inequalities are in the spirit of Brezis-Vázquez in the Euclidean spaces. As direct consequences, we establish several Hardy type inequalities that provide substantial improvements as well as simple understandings to many known Hardy inequalities and Hardy-Poincaré-Sobolev type inequalities on hyperbolic spaces in the literature.

An optimal improvement for the Hardy inequality on the hyperbolic space and related manifolds

Proceedings, 2019

We prove optimal improvements of the Hardy inequality on the hyperbolic space. Here, optimal means that the resulting operator is critical in the sense of , namely the associated inequality cannot be further improved. Such inequalities arise from more general, optimal ones valid for the operator P λ := -∆ H N -λ where 0 ≤ λ ≤ λ1(H N ) and λ1(H N ) is the bottom of the L 2 spectrum of -∆ H N , a problem that had been studied in only for the operator P λ 1 (H N ) . A different, critical and new inequality on H N , locally of Hardy type, is also shown. Such results have in fact greater generality since they are proved on general Cartan-Hadamard manifolds under curvature assumptions, possibly depending on the point. Existence/nonexistence of extremals for the related Hardy-Poincaré inequalities are also proved using concentration-compactness technique and a Liouville comparison theorem. As applications of our inequalities we obtain an improved Rellich inequality and we derive a quantitative version of Heisenberg-Pauli-Weyl uncertainty principle for the operator P λ . In the case of a Cartan-Hadamard manifold M of dimension N (namely, a manifold which is complete, simply-connected, and has everywhere non-positive sectional curvature), the geodesic distance function d(x, x 0 ), where x 0 ∈ M , satisfies all the assumptions of the weight ̺ and the above inequality holds with best constant N -2 2 2 , see . In particular, considering the most important example of Cartan-Hadamard manifold, namely the hyperbolic space H N , inequality (1.1) reads with r := d(x, x 0 ) and x 0 ∈ H N is a fixed pole.

Hardy–Sobolev inequalities and hyperbolic symmetry

Rendiconti Lincei-matematica E Applicazioni, 2008

Mathematical analysis. -Hardy-Sobolev inequalities and hyperbolic symmetry, by DANIELE CASTORINA, ISABELLA FABBRI, GIANNI MANCINI and KUNNATH SANDEEP, communicated on 12 June 2008. ABSTRACT. -We discuss uniqueness and nondegeneracy of extremals for some weighted Sobolev inequalities and give some applications to Grushin and scalar curvature type equations. The main theme is hyperbolic symmetry. KEY WORDS: Nonlinear PDE; hyperbolic symmetry; Hardy-Sobolev inequalities. MATHEMATICS SUBJECT CLASSIFICATION (2000): Primary 35J60; Secondary 35B05, 35A15.

Hardy-Sobolev inequalities, hyperbolic symmetry and the Webster scalar curvature problem

We discuss the problem u = (x)u p(t) jyjt ; u 2 D 1;2(RN ) where x = (y;z) 2 Rk Rh = RN; 2 k < N;t 2 (0; 2);p(t) := N 2t+2 N 2 in connection with Grushin-type equations and the Webster scalar curvature problem, providing various existence results. We highlight the role of hyperbolic symmetry in nondegeneracy and uniqueness questions and present a uniqueness result for semilinear elliptic equa- tions in Hyperbolic space which applies to the above equation when N > 3; = 1 and to semilinear Grushin-type equations as well.

Trudinger–Moser inequality in the hyperbolic space ℍN

Advances in Nonlinear Analysis, 2013

We prove a version of the Trudinger-Moser inequality in the hyperbolic space H N , which gives a sharper version of the Trudinger-Moser inequality on the Euclidean unit ball, as well as a hyperbolic space version of the Onofri inequality, and prove the existence of extremal functions to some related problems.

Parabolic weighted Sobolev–Poincaré type inequalities

2022

We derive weighted Sobolev-Poincaré type inequalities in function spaces concerned with parabolic partial differential equations. We consider general weights depending on both space and time variables belonging to a Muckenhoupt class, so-called the parabolic A p-class, where only the parabolic cubes are involved in the definition.

Uniform Poincaré inequalities on measured metric spaces

manuscripta mathematica

Consider a proper geodesic metric space (X, d) equipped with a Borel measure µ. We establish a family of uniform Poincaré inequalities on (X, d, µ) if it satisfies a local Poincaré inequality (P loc) and a condition on growth of volume. Consequently if µ is doubling and supports (P loc) then it satisfies a (σ, β, σ)-Poincaré inequality. If (X, d, µ) is a δ-hyperbolic space then using the volume comparison theorem in [3] we obtain a uniform Poincaré inequality with exponential growth of the Poincaré constant. Next we relate growth of Poincaré constants to growth of discrete subgroups of isometries of X which act on it properly. We show that if X is the universal cover of a compact CD(K, ∞) space then it supports a uniform Poincaré inequality and the Poincaré constant depends on the growth of the fundamental group.

On the extremal functions of Sobolev–Poincaré inequality

Pacific Journal of Mathematics, 2004

We prove the existence of extremal functions of Sobolev-Poincaré inequality on S n for p ∈ (1, (1 + √ 1 + 8n)/4). For general n-dimensional compact Riemannian manifolds embedded in R n+1 , such an existence result is proved for p ∈ (n/(n-1), (1 + √ 1 + 8n)/4). [9] M. Zhu, Sharp Poincaré-Sobolev inequalities and the shortest length of simple closed geodesics on a topological two sphere, preprint.

Improved Poincaré inequalities

Nonlinear Analysis: Theory, Methods & Applications, 2012

Although the Hardy inequality corresponding to one quadratic singularity, with optimal constant, does not admit any extremal function, it is well known that such a potential can be improved, in the sense that a positive term can be added to the quadratic singularity without violating the inequality, and even a whole asymptotic expansion can be build, with optimal constants for each term. This phenomenon has not been much studied for other inequalities. Our purpose is to prove that it also holds for the gaussian Poincaré inequality. The method is based on a recursion formula, which allows to identify the optimal constants in the asymptotic expansion, order by order. We also apply the same strategy to a family of Hardy-Poincaré inequalities which interpolate between Hardy and gaussian Poincaré inequalities.

Poincaré and mean value inequalities for hypersurfaces in Riemannian manifolds and applications

Asian Journal of Mathematics, 2017

In the first part of this paper we prove some new Poincaré inequalities, with explicit constants, for domains of any hypersurface of a Riemannian manifold with sectional curvatures bounded from above. This inequalities involve the first and the second symmetric functions of the eigenvalues of the second fundamental form of such hypersurface. We apply these inequalities to derive some isoperimetric inequalities and to estimate the volume of domains enclosed by compact self-shrinkers in terms of its scalar curvature. In the second part of the paper we prove some mean value inequalities and as consequences we derive some monotonicity results involving the integral of the mean curvature. ≤ C(p, Ω) ˆΩ |∇f | p dx 1 p. This is the Poincaré-Wirtinger inequality. An interesting question about these inequalities is to know the dependence of the Poincaré constant C(p, Ω) on the geometry of the domain Ω or H. Alencar was partially supported by CNPq of Brazil 2010 Mathematics Subject Classification. 53C21, 53C42.

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