scholarly journals Existence and non-existence of global solutions for semilinear heat equations and inequalities on sub-Riemannian manifolds, and Fujita exponent on unimodular Lie groups

2022 ◽  
Vol 308 ◽  
pp. 455-473
Author(s):  
Michael Ruzhansky ◽  
Nurgissa Yessirkegenov
Keyword(s):  
Lie Groups ◽  
Riemannian Manifolds ◽  
Global Solutions ◽  
Heat Equations

Bézier Curves on Riemannian Manifolds and Lie Groups With Kinematics Applications

1994 ◽  
Author(s):  
Frank C. Park ◽  
Bahram Ravani
Keyword(s):  
Riemannian Manifold ◽  
Rigid Body ◽  
Lie Groups ◽  
Riemannian Manifolds ◽  
Group Structure ◽  
Recursive Algorithm ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Spatial Displacements ◽  
Natural Way

Abstract In this article we generalize the concept of Bézier curves to curved spaces, and illustrate this generalization with an application in kinematics. We show how De Casteljau’s algorithm for constructing Bézier curves can be extended in a natural way to Riemannian manifolds. We then consider a special class of Riemannian manifold, the Lie groups. Because of their algebraic group structure Lie groups admit an elegant, efficient recursive algorithm for constructing Bézier curves. Spatial displacements of a rigid body also form a Lie group, and can therefore be interpolated (in the Bezier sense) using this recursive algorithm. We apply this algorithm to the kinematic problem of trajectory generation or motion interpolation for a moving rigid body.

scholarly journals r-Harmonic and Complex Isoparametric Functions on the Lie Groups $${{\mathbb {R}}}^m \ltimes {{\mathbb {R}}}^n$$ and $${{\mathbb {R}}}^m \ltimes \mathrm {H}^{2n+1}$$

2020 ◽  
Vol 58 (4) ◽  
pp. 477-496
Author(s):  
Sigmundur Gudmundsson ◽  
Marko Sobak
Keyword(s):  
Heisenberg Group ◽  
Lie Groups ◽  
Lie Group ◽  
Riemannian Manifolds ◽  
Harmonic Functions ◽  
Semidirect Products ◽  
Simply Connected ◽  
Isoparametric Functions ◽  
General Method

Abstract In this paper we introduce the notion of complex isoparametric functions on Riemannian manifolds. These are then employed to devise a general method for constructing proper r-harmonic functions. We then apply this to construct the first known explicit proper r-harmonic functions on the Lie group semidirect products $${{\mathbb {R}}}^m \ltimes {{\mathbb {R}}}^n$$ R m ⋉ R n and $${{\mathbb {R}}}^m \ltimes \mathrm {H}^{2n+1}$$ R m ⋉ H 2 n + 1 , where $$\mathrm {H}^{2n+1}$$ H 2 n + 1 denotes the classical $$(2n+1)$$ ( 2 n + 1 ) -dimensional Heisenberg group. In particular, we construct such examples on all the simply connected irreducible four-dimensional Lie groups.

scholarly journals Riemannian Means as Solutions of Variational Problems

2006 ◽  
Vol 9 ◽  
pp. 86-103 ◽  
Author(s):  
Luís Machado ◽  
F. Silva Leite ◽  
Knut Hüper
Keyword(s):  
Riemannian Manifold ◽  
Variational Problem ◽  
Lie Groups ◽  
Riemannian Manifolds ◽  
Single Point ◽  
Variational Problems ◽  
Compact Lie Groups ◽  
Data Set ◽  
The Given ◽  
Best Fit

We formulate a variational problem on a Riemannian manifoldMwhose solutions are piecewise smooth geodesies that best fit a given data set of time labelled points inM. By a limiting process, these solutions converge to a single point inM. which we prove to be the Riemannian mean of the given points for some particular Riemannian manifolds such as Euclidean spaces, connected and compact Lie groups, and spheres.

scholarly journals A Liouville Theorem for Superlinear Heat Equations on Riemannian Manifolds

2019 ◽  
Vol 87 (2) ◽  
pp. 303-313 ◽  
Author(s):  
Daniele Castorina ◽  
Carlo Mantegazza ◽  
Berardino Sciunzi
Keyword(s):  
Riemannian Manifolds ◽  
Liouville Theorem ◽  
Heat Equations

scholarly journals Ancient solutions of superlinear heat equations on Riemannian manifolds

2020 ◽  
pp. 2050033
Author(s):  
Daniele Castorina ◽  
Carlo Mantegazza
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Heat Equations ◽  
Qualitative Properties ◽  
Ancient Solutions

We study some qualitative properties of ancient solutions of superlinear heat equations on a Riemannian manifold, with particular interest in positivity and constancy in space.

scholarly journals Global solutions of semilinear heat equations in Hilbert spaces

1996 ◽  
Vol 1 (3) ◽  
pp. 263-276 ◽  
Author(s):  
G. Mihai Iancu ◽  
M. W. Wong
Keyword(s):  
Asymptotic Behavior ◽  
Hilbert Spaces ◽  
Differential Operators ◽  
Global Solutions ◽  
Linear Operators ◽  
Heat Equations ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Strongly Continuous Semigroups ◽  
Pseudo Differential Operators

The existence, uniqueness, regularity and asymptotic behavior of global solutions of semilinear heat equations in Hilbert spaces are studied by developing new results in the theory of one-parameter strongly continuous semigroups of bounded linear operators. Applications to special semilinear heat equations inL 2(ℝn)governed by pseudo-differential operators are given.

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