Gagliardo-Nirenberg Inequalities for Horizontal Vector Fields in the Engel Group and in the Seven-Dimensional Quaternionic Heisenberg Group

2015 ◽  
pp. 287-312 ◽  
Author(s):  
Annalisa Baldi ◽  
Bruno Franchi ◽  
Francesca Tripaldi
Keyword(s):  
Heisenberg Group ◽  
Vector Fields ◽  
Engel Group ◽  
Quaternionic Heisenberg Group ◽  
Horizontal Vector

DIV–CURL TYPE THEOREM, H-CONVERGENCE AND STOKES FORMULA IN THE HEISENBERG GROUP

2006 ◽  
Vol 08 (01) ◽  
pp. 67-99 ◽  
Author(s):  
BRUNO FRANCHI ◽  
NICOLETTA TCHOU ◽  
MARIA CARLA TESI
Keyword(s):  
Heisenberg Group ◽  
Vector Fields ◽  
Differential Operators ◽  
Type Theorem ◽  
Divergence Form ◽  
Second Order ◽  
Order Differential Operator ◽  
Contact Manifolds ◽  
Stokes Formula ◽  
Horizontal Vector

In this paper, we prove a div–curl type theorem in the Heisenberg group ℍ1, and then we develop a theory of H-convergence for second order differential operators in divergence form in ℍ1. The div–curl theorem requires an intrinsic notion of the curl operator in ℍ1 (that we denote by curlℍ), that turns out to be a second order differential operator in the left invariant horizontal vector fields. As an evidence of the coherence of this definition, we prove an intrinsic Stokes formula for curlℍ. Eventually, we show that this notion is related to one of the exterior differentials in Rumin's complex on contact manifolds.

Semilinear PDEs in the Heisenberg group: The role of the right invariant vector fields

2010 ◽  
Vol 72 (2) ◽  
pp. 987-997 ◽  
Author(s):  
Isabeau Birindelli ◽  
Fausto Ferrari ◽  
Enrico Valdinoci
Keyword(s):  
Heisenberg Group ◽  
Vector Fields ◽  
Invariant Vector ◽  
Semilinear Pdes ◽  
The Right ◽  
Invariant Vector Fields

scholarly journals High Order Fefferman-Phong Type Inequalities in Carnot Groups and Regularity for Degenerate Elliptic Operators plus a Potential

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Pengcheng Niu ◽  
Kelei Zhang
Keyword(s):  
Euclidean Space ◽  
Vector Fields ◽  
Carnot Group ◽  
A Priori ◽  
Elliptic Operators ◽  
High Order ◽  
Carnot Groups ◽  
Degenerate Elliptic Operators ◽  
Degenerate Elliptic ◽  
Horizontal Vector

Let{X1,X2,…,Xm}be the basis of space of horizontal vector fields in a Carnot groupG=(Rn;∘) (m<n). We prove high order Fefferman-Phong type inequalities inG. As applications, we derive a prioriLp(G)estimates for the nondivergence degenerate elliptic operatorsL=-∑i,j=1maij(x)XiXj+V(x)withVMOcoefficients and a potentialVbelonging to an appropriate Stummel type class introduced in this paper. Some of our results are also new even for the usual Euclidean space.

scholarly journals Nonexistence Results for Semilinear Equations in Carnot Groups

2013 ◽  
Vol 1 ◽  
pp. 130-146 ◽  
Author(s):  
Fausto Ferrari ◽  
Andrea Pinamonti
Keyword(s):  
Heisenberg Group ◽  
Carnot Group ◽  
Carnot Groups ◽  
Semilinear Equations ◽  
Engel Group ◽  
Large Step

Abstract In this paper, following [3], we provide some nonexistence results for semilinear equations in the the class of Carnot groups of type ★.This class, see [20], contains, in particular, all groups of step 2; like the Heisenberg group, and also Carnot groups of arbitrarly large step. Moreover, we prove some nonexistence results for semilinear equations in the Engel group, which is the simplest Carnot group that is not of type ★.

scholarly journals CONJUGATE POINTS ON THE QUATERNIONIC HEISENBERG GROUP

2003 ◽  
Vol 40 (1) ◽  
pp. 61-72 ◽  
Author(s):  
Chang-Rim Jang ◽  
Jun-Kon Kim ◽  
Yeon-Wook Kim ◽  
Keun Park
Keyword(s):  
Heisenberg Group ◽  
Conjugate Points ◽  
Quaternionic Heisenberg Group

scholarly journals On Lipschitz vector fields and the Cauchy problem in homogeneous groups

2018 ◽  
Vol 20 (05) ◽  
pp. 1750057
Author(s):  
Valentino Magnani ◽  
Dario Trevisan
Keyword(s):  
Cauchy Problem ◽  
Vector Fields ◽  
Equilibrium Points ◽  
Uniqueness Result ◽  
Well Posedness ◽  
Homogeneous Groups ◽  
The Cauchy Problem ◽  
Horizontal Vector

We introduce a class of “Lipschitz horizontal” vector fields in homogeneous groups, for which we show equivalent descriptions, e.g., in terms of suitable derivations. We then investigate the associated Cauchy problem, providing a uniqueness result both at equilibrium points and for vector fields of an involutive submodule of Lipschitz horizontal vector fields. We finally exhibit a counterexample to the general well-posedness theory for Lipschitz horizontal vector fields, in contrast with the Euclidean theory.

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