scholarly journals Some remarks on the Sobolev inequality in Riemannian manifolds

2021 ◽  
Author(s):  
Daniele Andreucci ◽  
Anatoli Tedeev
Keyword(s):  
Riemannian Manifolds ◽  
Sobolev Inequality

scholarly journals Solving Liouville-type Problems on Manifolds with Poincaré-Sobolev Inequality by Broadening q-Energy from Finite to Infinite

2017 ◽  
Vol 9 (4) ◽  
pp. 1
Author(s):  
Lina Wu
Keyword(s):  
Riemannian Manifolds ◽  
Sobolev Inequality ◽  
Differential Forms ◽  
Computational Method ◽  
Liouville Type ◽  
Balanced Growth ◽  
Curved Manifolds ◽  
Research Findings ◽  
Liouville Type Results ◽  
Flat Manifolds

The aim of this article is to investigate Liouville-type problems on complete non-compact Riemannian manifolds with Poincaré-Sobolev Inequality. Two significant technical breakthroughs are demonstrated in research findings. The first breakthrough is an extension from non-flat manifolds with non-negative Ricci curvatures to curved manifolds with Ricci curvatures varying among negative values, zero, and positive values. Poincaré-Sobolev Inequality has been applied to overcome difficulties of an extension on manifolds. Poincaré-Sobolev Inequality has offered a special structure on curved manifolds with a mix of Ricci curvature signs. The second breakthrough is a generalization of $q$-energy from finite to infinite. At this point, a technique of $p$-balanced growth has been introduced to overcome difficulties of broadening from finite $q$-energy in $L^q$ spaces to infinite $q$-energy in non-$L^q$ spaces. An innovative computational method and new estimation techniques are illustrated. At the end of this article, Liouville-type results including vanishing properties for differential forms and constancy properties for differential maps have been verified on manifolds with Poincaré-Sobolev Inequality approaching to infinite $q$-energy growth.

scholarly journals A sharp Sobolev inequality on Riemannian manifolds

2003 ◽  
Vol 2 (1) ◽  
pp. 1-31 ◽  
Author(s):  
YanYan Li ◽  
◽  
Tonia Ricciardi ◽  
Keyword(s):  
Riemannian Manifolds ◽  
Sobolev Inequality

scholarly journals A sharp Sobolev inequality on Riemannian manifolds

2002 ◽  
Vol 335 (6) ◽  
pp. 519-524 ◽  
Author(s):  
Yan Yan Li ◽  
Tonia Ricciardi
Keyword(s):  
Riemannian Manifolds ◽  
Sobolev Inequality

Homogeneous sharp Sobolev inequalities on product manifolds

2006 ◽  
Vol 136 (2) ◽  
pp. 277-300 ◽  
Author(s):  
Jurandir Ceccon ◽  
Marcos Montenegro
Keyword(s):  
Riemannian Manifolds ◽  
Sobolev Inequality ◽  
Sobolev Inequalities ◽  
Extremal Functions ◽  
Mass Transportation ◽  
Best Constant ◽  
First Order ◽  
Image Position ◽  
Optimal Sobolev Inequality ◽  
Homogeneous Convex

Let (M, g) and (N, h) be compact Riemannian manifolds of dimensions m and n, respectively. For p-homogeneous convex functions f(s, t) on [0,∞) × [0, ∞), we study the validity and non-validity of the first-order optimal Sobolev inequality on H1, p(M × N) where and Kf = Kf (m, n, p) is the best constant of the homogeneous Sobolev inequality on D1, p (Rm+n), The proof of the non-validity relies on the knowledge of extremal functions associated with the Sobolev inequality above. In order to obtain such extremals we use mass transportation and convex analysis results. Since variational arguments do not work for general functions f, we investigate the validity in a uniform sense on f and argue with suitable approximations of f which are also essential in the non-validity. Homogeneous Sobolev inequalities on product manifolds are connected to elliptic problems involving a general class of operators.

Rough isometry and energy-finite solutions for the Schrödinger operator on Riemannian manifolds

2003 ◽  
Vol 133 (4) ◽  
pp. 855-873 ◽  
Author(s):  
Seok Woo Kim ◽  
Yong Hah Lee
Keyword(s):  
Riemannian Manifolds ◽  
Schrödinger Operator ◽  
Sobolev Inequality ◽  
Harmonic Functions ◽  
Schrodinger Operator ◽  
Dirichlet Integral ◽  
Local Conditions ◽  
Complete Riemannian Manifolds ◽  
Bounded Harmonic Functions ◽  
Rough Isometry

In this paper, we prove that the dimension of the space of bounded energy-finite solutions for the Schrödinger operator is invariant under rough isometries between complete Riemannian manifolds satisfying the local volume condition, the local Poincaré inequality and the local Sobolev inequality. We also prove that the dimension of the space of bounded harmonic functions with finite Dirichlet integral is invariant under rough isometries between complete Riemannian manifolds satisfying the same local conditions. These results generalize those of Kanai, Grigor'yan, the second author, and Li and Tam.

On the equicontinuity of families of inverse mappings of Riemannian manifolds

2019 ◽  
Vol 16 (4) ◽  
pp. 557-566
Author(s):  
Denis Ilyutko ◽  
Evgenii Sevost'yanov
Keyword(s):  
Riemannian Manifolds ◽  
Type Inequality ◽  
The Given ◽  
Range Of Values

We study homeomorphisms of Riemannian manifolds with unbounded characteristic such that the inverse mappings satisfy the Poletsky-type inequality. It is established that their families are equicontinuous if the function Q which is related to the Poletsky inequality and is responsible for a distortion of the modulus, is integrable in the given domain, here the original manifold is connected and the domain of definition and the range of values of mappings have compact closures.

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