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.

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.

scholarly journals A Korn-type inequality in SBD for functions with small jump sets

2017 ◽  
Vol 27 (13) ◽  
pp. 2461-2484 ◽  
Author(s):  
Manuel Friedrich
Keyword(s):  
Bounded Variation ◽  
Elastic Strain ◽  
Special Functions ◽  
Type Inequality ◽  
Rigid Motion ◽  
Exceptional Set ◽  
Finite Perimeter ◽  
Functions Of Bounded Deformation ◽  
Jump Set

We present a Korn-type inequality in a planar setting for special functions of bounded deformation. We prove that for each function in [Formula: see text] with a sufficiently small jump set the distance of the function and its derivative from an infinitesimal rigid motion can be controlled in terms of the linearized elastic strain outside of a small exceptional set of finite perimeter. Particularly, the result shows that each function in [Formula: see text] has bounded variation away from an arbitrarily small part of the domain.

scholarly journals Maximal function characterizations of Hardy spaces associated to homogeneous higher order elliptic operators

2016 ◽  
Vol 28 (5) ◽  
pp. 823-856 ◽  
Author(s):  
Jun Cao ◽  
Svitlana Mayboroda ◽  
Dachun Yang
Keyword(s):  
Elliptic Operator ◽  
Maximal Function ◽  
Hardy Spaces ◽  
Divergence Form ◽  
Higher Order ◽  
Square Functions ◽  
Maximal Interval ◽  
Measurable Coefficients ◽  
Order Elliptic Operator ◽  
Higher Order Elliptic Operator

AbstractLet L be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients and ${(p_{-}(L),p_{+}(L))}$ be the maximal interval of exponents ${q\in[1,\infty]}$ such that the semigroup ${\{e^{-tL}\}_{t>0}}$ is bounded on ${L^{q}(\mathbb{R}^{n})}$. In this article, the authors establish the non-tangential maximal function characterizations of the associated Hardy spaces ${H_{L}^{p}(\mathbb{R}^{n})}$ for all ${p\in(0,p_{+}(L))}$, which when ${p=1}$, answers a question asked by Deng, Ding and Yao in [21]. Moreover, the authors characterize ${H_{L}^{p}(\mathbb{R}^{n})}$ via various versions of square functions and Lusin-area functions associated to the operator L.

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 New Estimations of Hermite–Hadamard Type Integral Inequalities for Special Functions

2021 ◽  
Vol 5 (4) ◽  
pp. 144
Author(s):  
Hijaz Ahmad ◽  
Muhammad Tariq ◽  
Soubhagya Kumar Sahoo ◽  
Jamel Baili ◽  
Clemente Cesarano
Keyword(s):  
Special Functions ◽  
Type Inequality ◽  
Integral Inequalities ◽  
The Novel ◽  
Current Pattern ◽  
Hadamard Inequality ◽  
Special Means ◽  
Algebraic Properties ◽  
Type Integral ◽  
Generalized Integral Inequalities

In this paper, we propose some generalized integral inequalities of the Raina type depicting the Mittag–Leffler function. We introduce and explore the idea of generalized s-type convex function of Raina type. Based on this, we discuss its algebraic properties and establish the novel version of Hermite–Hadamard inequality. Furthermore, to improve our results, we explore two new equalities, and employing these we present some refinements of the Hermite–Hadamard-type inequality. A few remarkable cases are discussed, which can be seen as valuable applications. Applications of some of our presented results to special means are given as well. An endeavor is made to introduce an almost thorough rundown of references concerning the Mittag–Leffler functions and the Raina functions to make the readers acquainted with the current pattern of emerging research in various fields including Mittag–Leffler and Raina type functions. Results established in this paper can be viewed as a significant improvement of previously known results.

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.

On Variational Inequalities Driven by Elliptic Operators Not in Divergence Form

2012 ◽  
Vol 12 (3) ◽  
Author(s):  
Michele Matzeu ◽  
Raffaella Servadei
Keyword(s):  
Variational Inequalities ◽  
Elliptic Operator ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Divergence Form ◽  
Elliptic Operators

AbstractIn this paper we study semilinear variational inequalities driven by an elliptic operator not in divergence form modeled bywhere Ω is a bounded domain of RN, N ≥ 3, with smooth boundary, A is the elliptic operator, not in divergence form, given byHere a

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