scholarly journals Overview and comparative analysis of the properties of the Hodge-de Rham and Tachibana operators

Filomat ◽  
2015 ◽  
Vol 29 (10) ◽  
pp. 2429-2436
Author(s):  
S.E. Stepanov ◽  
I.I. Tsyganok ◽  
J. Mikes
Keyword(s):  
Comparative Analysis ◽  
Riemannian Manifolds ◽  
Differential Forms ◽  
Second Order ◽  
Exterior Differential ◽  
Basic Properties ◽  
De Rham

In the present paper we consider two natural, elliptic, self-adjoint second order di_erential operators acting on exterior differential forms on Riemannian manifolds. These operators are the wellknown Hodge-de Rham and little-known Tachibana operators. Basic properties of these operators are very similar, or vice versa are dual with respect to each other. We review the results (partly obtained by the authors) on the geometry of these operators and demonstrate the comparative analysis of their properties.

Boundary value problems for differential forms on compact Riemannian manifolds, Part II

Analysis ◽  
2007 ◽  
Vol 27 (4) ◽  
Author(s):  
Jürgen Bolik
Keyword(s):  
Boundary Value Problems ◽  
Riemannian Manifolds ◽  
Boundary Value ◽  
Differential Forms ◽  
Second Order ◽  
First Order

SummaryThis paper provides solutions to second order boundary value problems for differential forms by means of the method applied in [3] for first order problems. These

ABSTRACT AND CLASSICAL HODGE–DE RHAM THEORY

2012 ◽  
Vol 10 (01) ◽  
pp. 91-111 ◽  
Author(s):  
NAT SMALE ◽  
STEVE SMALE
Keyword(s):  
Riemannian Manifolds ◽  
Metric Spaces ◽  
Differential Forms ◽  
Small Scale ◽  
Smooth Functions ◽  
Compact Metric Spaces ◽  
Rham Complex ◽  
De Rham Theory ◽  
De Rham ◽  
Chain Maps

In previous work, with Bartholdi and Schick [1], the authors developed a Hodge–de Rham theory for compact metric spaces, which defined a cohomology of the space at a scale α. Here, in the case of Riemannian manifolds at a small scale, we construct explicit chain maps between the de Rham complex of differential forms and the L2 complex at scale α, which induce isomorphisms on cohomology. We also give estimates that show that on smooth functions, the Laplacian of [1], when appropriately scaled, is a good approximation of the classical Laplacian.

scholarly journals A Note on the Strong Maximum Principle for Fully Nonlinear Equations on Riemannian Manifolds

2021 ◽  
Author(s):  
Alessandro Goffi ◽  
Francesco Pediconi
Keyword(s):  
Maximum Principle ◽  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Strong Maximum Principle ◽  
Second Order ◽  
Fully Nonlinear Equations ◽  
Nonlinear Operators ◽  
Fully Nonlinear ◽  
Second Order Equations ◽  
Curvature Type

AbstractWe investigate strong maximum (and minimum) principles for fully nonlinear second-order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci’s extremal operators, some singular operators such as those modeled on the p- and $$\infty $$ ∞ -Laplacian, and mean curvature-type problems. As a byproduct, we establish new strong comparison principles for some second-order uniformly elliptic problems when the manifold has nonnegative sectional curvature.

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 The de Rham Cohomology through Hilbert Space Methods

2019 ◽  
pp. 455-465
Author(s):  
Ihsane Malass ◽  
Nikolai Tarkhanov
Keyword(s):  
Hilbert Space ◽  
Elliptic Boundary ◽  
Differential Forms ◽  
Adjoint Operator ◽  
Lagrange Equations ◽  
De Rham Cohomology ◽  
Elliptic Boundary Value ◽  
De Rham ◽  
Canonical Representations ◽  
The Neumann Problem

We discuss canonical representations of the de Rham cohomology on a compact manifold with boundary. They are obtained by minimising the energy integral in a Hilbert space of differential forms that belong along with the exterior derivative to the domain of the adjoint operator. The corresponding Euler- Lagrange equations reduce to an elliptic boundary value problem on the manifold, which is usually referred to as the Neumann problem after Spencer

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