Tangent Halfspaces to Sets of Finite Perimeter in Carnot Groups

2009 ◽  
pp. 1-16 ◽  
Author(s):  
Luigi Ambrosio
Keyword(s):  
Carnot Groups ◽  
Sets Of Finite Perimeter ◽  
Finite Perimeter

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 Rectifiability of Sets of Finite Perimeter in Carnot Groups: Existence of a Tangent Hyperplane

2009 ◽  
Vol 19 (3) ◽  
pp. 509-540 ◽  
Author(s):  
Luigi Ambrosio ◽  
Bruce Kleiner ◽  
Enrico Le Donne
Keyword(s):  
Carnot Groups ◽  
Tangent Hyperplane ◽  
Sets Of Finite Perimeter ◽  
Finite Perimeter

BV Functions and Sets of Finite Perimeter

2018 ◽  
pp. 166-226
Author(s):  
Lawrence C. Evans ◽  
Ronald F. Garzepy
Keyword(s):  
Sets Of Finite Perimeter ◽  
Finite Perimeter ◽  
Bv Functions

Embeddings for the space on sets of finite perimeter

2019 ◽  
Vol 150 (5) ◽  
pp. 2442-2461 ◽  
Author(s):  
Nikolai V. Chemetov ◽  
Anna L. Mazzucato
Keyword(s):  
Incompressible Fluid ◽  
Viscous Incompressible Fluid ◽  
Rigid Bodies ◽  
Deformation Tensor ◽  
Integrable Deformation ◽  
Sets Of Finite Perimeter ◽  
Open Set ◽  
Finite Perimeter ◽  
Continuous Embedding

AbstractGiven an open set with finite perimeter $\Omega \subset {\open R}^n$, we consider the space $LD_\gamma ^{p}(\Omega )$, $1\les p<\infty $, of functions with pth-integrable deformation tensor on Ω and with pth-integrable trace value on the essential boundary of Ω. We establish the continuous embedding $LD_\gamma ^{p}(\Omega )\subset L^{pN/(N-1)}(\Omega )$. The space $LD_\gamma ^{p}(\Omega )$ and this embedding arise naturally in studying the motion of rigid bodies in a viscous, incompressible fluid.

scholarly journals On locally essentially bounded divergence measure fields and sets of locally finite perimeter

2020 ◽  
Vol 13 (2) ◽  
pp. 179-217 ◽  
Author(s):  
Giovanni E. Comi ◽  
Kevin R. Payne
Keyword(s):  
Approximation Theory ◽  
Smooth Boundary ◽  
Direct Proof ◽  
Leibniz Rule ◽  
Divergence Measure ◽  
Locally Finite ◽  
Sets Of Finite Perimeter ◽  
Finite Perimeter ◽  
Approximation Lemma ◽  
Natural Way

AbstractChen, Torres and Ziemer ([9], 2009) proved the validity of generalized Gauss–Green formulas and obtained the existence of interior and exterior normal traces for essentially bounded divergence measure fields on sets of finite perimeter using an approximation theory through sets with a smooth boundary. However, it is known that the proof of a crucial approximation lemma contained a gap. Taking inspiration from a previous work of Chen and Torres ([7], 2005) and exploiting ideas of Vol’pert ([29], 1985) for essentially bounded fields with components of bounded variation, we present here a direct proof of generalized Gauss–Green formulas for essentially bounded divergence measure fields on sets of finite perimeter which includes the existence and essential boundedness of the normal traces. Our approach appears to be simpler since it does not require any special approximation theory for the domains and it relies only on the Leibniz rule for divergence measure fields. This freedom allows one to localize the constructions and to derive more general statements in a natural way.

scholarly journals ON THE MEASURE-THEORETIC FOUNDATIONS OF THE SECOND LAW OF THERMODYNAMICS

2002 ◽  
Vol 12 (05) ◽  
pp. 721-736 ◽  
Author(s):  
ALFREDO MARZOCCHI ◽  
ALESSANDRO MUSESTI
Keyword(s):  
Measure Theory ◽  
Geometric Measure Theory ◽  
Second Law Of Thermodynamics ◽  
Second Law ◽  
Weak Version ◽  
Balance Laws ◽  
Geometric Measure ◽  
Sets Of Finite Perimeter ◽  
Finite Perimeter ◽  
Theoretic Foundations

Balance laws of the type of entropy are treated in the framework of geometric measure theory, and a weak version, although conceptually simple, of the Second Law of Thermodynamics is introduced, allowing extensions to measure-valued entropy productions and to sets of finite perimeter as subbodies.

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