On the structure of finite perimeter sets in step 2 Carnot groups

2003 ◽  
Vol 13 (3) ◽  
pp. 421-466 ◽  
Author(s):  
Bruno Franchi ◽  
Raul Serapioni ◽  
Francesco Serra Cassano
Keyword(s):  
Carnot Groups ◽  
Finite Perimeter ◽  
Finite Perimeter Sets

scholarly journals A SHORT NOTE ON ENHANCED DENSITY SETS

2011 ◽  
Vol 53 (3) ◽  
pp. 631-635 ◽  
Author(s):  
SILVANO DELLADIO
Keyword(s):  
Simple Proof ◽  
Short Note ◽  
Manuscripta Math ◽  
Locally Finite ◽  
Finite Perimeter ◽  
Measurable Set ◽  
Finite Perimeter Sets

AbstractWe give a simple proof of a statement extending Fu's (J.H.G. Fu, Erratum to ‘some remarks on legendrian rectiable currents’, Manuscripta Math. 113(3) (2004), 397–401) result: ‘If Ω is a set of locally finite perimeter in ℝ2, then there is no function f ∈ C1(ℝ2) such that ∇f(x1, x2) = (x2, 0) at a.e. (x1, x2) ∈ Ω’. We also prove that every measurable set can be approximated arbitrarily closely in L1 by subsets that do not contain enhanced density points. Finally, we provide a new proof of a Poincaré-type lemma for locally finite perimeter sets, which was first stated by Delladio (S. Delladio, Functions of class C1 subject to a Legendre condition in an enhanced density set, to appear in Rev. Mat. Iberoamericana).

scholarly journals Semigenerated Carnot algebras and applications to sub-Riemannian perimeter

2021 ◽  
Author(s):  
Enrico Le Donne ◽  
Terhi Moisala
Keyword(s):  
Half Space ◽  
Complete Characterization ◽  
Carnot Groups ◽  
Call Type ◽  
New Class ◽  
Sets Of Finite Perimeter ◽  
Finite Perimeter

AbstractThis paper contributes to the study of sets of finite intrinsic perimeter in Carnot groups. Our intent is to characterize in which groups the only sets with constant intrinsic normal are the vertical half-spaces. Our viewpoint is algebraic: such a phenomenon happens if and only if the semigroup generated by each horizontal half-space is a vertical half-space. We call semigenerated those Carnot groups with this property. For Carnot groups of nilpotency step 3 we provide a complete characterization of semigeneration in terms of whether such groups do not have any Engel-type quotients. Engel-type groups, which are introduced here, are the minimal (in terms of quotients) counterexamples. In addition, we give some sufficient criteria for semigeneration of Carnot groups of arbitrary step. For doing this, we define a new class of Carnot groups, which we call type $$(\Diamond )$$ ( ◊ ) and which generalizes the previous notion of type $$(\star )$$ ( ⋆ ) defined by M. Marchi. As an application, we get that in type $$ (\Diamond ) $$ ( ◊ ) groups and in step 3 groups that do not have any Engel-type algebra as a quotient, one achieves a strong rectifiability result for sets of finite perimeter in the sense of Franchi, Serapioni, and Serra-Cassano.

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.

Extremals in some classes of Carnot groups

2011 ◽  
Vol 55 (3) ◽  
pp. 633-646 ◽  
Author(s):  
TiRen Huang ◽  
XiaoPing Yang
Keyword(s):  
Carnot Groups
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