SOME -HARDY AND -RELLICH TYPE INEQUALITIES WITH REMAINDER TERMS

2021 ◽  
pp. 1-20
Author(s):  
YONGYANG JIN ◽  
SHOUFENG SHEN
Keyword(s):  
Riemannian Manifolds ◽  
Bounded Domains ◽  
Hadamard Manifolds ◽  
Sharp Constants ◽  
Hyperbolic Functions

Abstract In this paper we obtain some improved $L^p$ -Hardy and $L^p$ -Rellich inequalities on bounded domains of Riemannian manifolds. For Cartan–Hadamard manifolds we prove the inequalities with sharp constants and with weights being hyperbolic functions of the Riemannian distance.

scholarly journals Quantitative Rellich inequalities on Finsler—Hadamard manifolds

2016 ◽  
Vol 18 (06) ◽  
pp. 1650020 ◽  
Author(s):  
Alexandru Kristály ◽  
Dušan Repovš
Keyword(s):  
Riemannian Manifolds ◽  
Negative Curvature ◽  
Finsler Geometry ◽  
Flag Curvature ◽  
Hadamard Manifolds ◽  
Hardy Inequalities

In this paper, we are dealing with quantitative Rellich inequalities on Finsler–Hadamard manifolds where the remainder terms are expressed by means of the flag curvature. By exploring various arguments from Finsler geometry and PDEs on manifolds, we show that more weighty curvature implies more powerful improvements in Rellich inequalities. The sharpness of the involved constants is also studied. Our results complement those of Yang, Su and Kong [Hardy inequalities on Riemannian manifolds with negative curvature, Commun. Contemp. Math. 16 (2014), Article ID: 1350043, 24 pp.].

scholarly journals Generalized Geodesic Convexity on Riemannian Manifolds

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 547 ◽  
Author(s):  
Izhar Ahmad ◽  
Meraj Ali Khan ◽  
Amira A. Ishan
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Global Minimum ◽  
Mean Value ◽  
Mathematical Programming Problem ◽  
Hadamard Manifolds ◽  
Geodesic Convexity ◽  
Invex Functions ◽  
Mean Value Inequality ◽  
Global Minimum Point

We introduce log-preinvex and log-invex functions on a Riemannian manifold. Some properties and relationships of these functions are discussed. A characterization for the existence of a global minimum point of a mathematical programming problem is presented. Moreover, a mean value inequality under geodesic log-preinvexity is extended to Cartan-Hadamard manifolds.

scholarly journals Estimates for Eigenvalues of the Elliptic Operator in Divergence Form on Riemannian Manifolds

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Shenyang Tan ◽  
Tiren Huang ◽  
Wenbin Zhang
Keyword(s):  
Elliptic Operator ◽  
Riemannian Manifolds ◽  
Special Functions ◽  
Type Inequality ◽  
Divergence Form ◽  
Hadamard Manifolds ◽  
Complete Submanifolds ◽  
Universal Inequalities ◽  
Complete Manifolds ◽  
Bounded Below

We investigate the Dirichlet weighted eigenvalue problem of the elliptic operator in divergence form on compact Riemannian manifolds(M,g,e-ϕdv). We establish a Yang-type inequality of this problem. We also get universal inequalities for eigenvalues of elliptic operators in divergence form on compact domains of complete submanifolds admitting special functions which include the Hadamard manifolds with Ricci curvature bounded below and any complete manifolds admitting eigenmaps to a sphere.

Best Constants for Moser-Trudinger Inequalities on High Dimensional Hyperbolic Spaces

2013 ◽  
Vol 13 (4) ◽  
Author(s):  
Guozhen Lu ◽  
Hanli Tang
Keyword(s):  
Riemannian Manifolds ◽  
Best Constants ◽  
Hyperbolic Spaces ◽  
High Dimensional ◽  
Sharp Constant ◽  
Heisenberg Groups ◽  
Euclidean Spaces ◽  
Sharp Constants ◽  
Trudinger Inequality

AbstractThough there have been extensive works on best constants for Moser-Trudinger inequalities in Euclidean spaces, Heisenberg groups or compact Riemannian manifolds, much less is known for sharp constants for the Moser-Trudinger inequalities on hyperbolic spaces. Earlier works only include the sharp constant for the Moser-Trudinger inequality on the twodimensional hyperbolic disc. In this paper, we establish best constants for several types of Moser-Trudinger inequalities on high dimensional hyperbolic spaces ℍ

HARDY INEQUALITIES ON RIEMANNIAN MANIFOLDS WITH NEGATIVE CURVATURE

2014 ◽  
Vol 16 (02) ◽  
pp. 1350043 ◽  
Author(s):  
QIAOHUA YANG ◽  
DAN SU ◽  
YINYING KONG
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Negative Curvature ◽  
Geodesic Distance ◽  
Euclidean Spaces ◽  
Sharp Constants ◽  
Hardy Inequalities ◽  
Simply Connected

Let M be a complete, simply connected Riemannian manifold with negative curvature. We obtain the sharp constants of Hardy and Rellich inequalities related to the geodesic distance on M. Furthermore, if M is with strictly negative curvature, we show that the LpHardy inequalities can be globally refined by adding remainder terms like the Brezis–Vázquez improvement in case p ≥ 2, which is contrary to the case of Euclidean spaces.

scholarly journals Multipolar Hardy inequalities on Riemannian manifolds

2018 ◽  
Vol 24 (2) ◽  
pp. 551-567 ◽  
Author(s):  
Francesca Faraci ◽  
Csaba Farkas ◽  
Alexandru Kristály
Keyword(s):  
Variational Methods ◽  
Riemannian Manifolds ◽  
Quantitative Information ◽  
Beltrami Operator ◽  
Hadamard Manifolds ◽  
Hardy Inequalities ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Complete Riemannian Manifolds

We prove multipolar Hardy inequalities on complete Riemannian manifolds, providing various curved counterparts of some Euclidean multipolar inequalities due to Cazacu and Zuazua [Improved multipolar Hardy inequalities, 2013]. We notice that our inequalities deeply depend on the curvature, providing (quantitative) information about the deflection from the flat case. By using these inequalities together with variational methods and group-theoretical arguments, we also establish non-existence, existence and multiplicity results for certain Schrödinger-type problems involving the Laplace-Beltrami operator and bipolar potentials on Cartan-Hadamard manifolds and on the open upper hemisphere, respectively.

scholarly journals Generalized sharp Hardy type and Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds

2009 ◽  
Vol 40 (4) ◽  
pp. 401-413 ◽  
Author(s):  
Shihshu Walter Wei ◽  
Ye Li
Keyword(s):  
Fixed Point ◽  
Euclidean Space ◽  
Riemannian Manifolds ◽  
Compact Support ◽  
Hardy’S Inequality ◽  
Sharp Constants ◽  
Hardy Type ◽  
Hardy's Inequality ◽  
Nirenberg Inequality

We prove generalized Hardy's type inequalities with sharp constants and Caffarelli-Kohn-Nirenberg inequalities with sharp constants on Riemannian manifolds $M$. When the manifold is Euclidean space we recapture the sharp Caffarelli-Kohn-Nirenberg inequality. By using a double limiting argument, we obtain an inequality that implies a sharp Hardy's inequality, for functions with compact support on the manifold $M $ (that is, not necessarily on a punctured manifold $ M \backslash \{ x_0 \} $ where $x_0$ is a fixed point in $M$). Some topological and geometric applications are discussed.

scholarly journals Sharp Caffarelli–Kohn–Nirenberg inequalities on Riemannian manifolds: the influence of curvature

2021 ◽  
pp. 1-26
Author(s):  
Van Hoang Nguyen
Keyword(s):  
Euclidean Space ◽  
Riemannian Manifolds ◽  
Remainder Term ◽  
Sharp Constant ◽  
Hadamard Manifolds ◽  
Best Constant ◽  
Quantitative Version ◽  
Rigidity Results ◽  
Ckn Inequalities ◽  
Complete Riemannian Manifolds

We first establish a family of sharp Caffarelli–Kohn–Nirenberg type inequalities (shortly, sharp CKN inequalities) on the Euclidean spaces and then extend them to the setting of Cartan–Hadamard manifolds with the same best constant. The quantitative version of these inequalities also is proved by adding a non-negative remainder term in terms of the sectional curvature of manifolds. We next prove several rigidity results for complete Riemannian manifolds supporting the Caffarelli–Kohn–Nirenberg type inequalities with the same sharp constant as in the Euclidean space of the same dimension. Our results illustrate the influence of curvature to the sharp CKN inequalities on the Riemannian manifolds. They extend recent results of Kristály (J. Math. Pures Appl. 119 (2018), 326–346) to a larger class of the sharp CKN inequalities.

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