Characterizations of sets of finite perimeter using the Ornstein-Uhlenbeck semigroup in the Gauss space

2022 ◽  
Vol 174 ◽  
pp. 103090
Author(s):  
Xiangyun Xie ◽  
Yu Liu ◽  
Haihui Wang
Keyword(s):  
Sets Of Finite Perimeter ◽  
Finite Perimeter ◽  
Gauss Space

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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