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'''Definition 1'''
 
'''Definition 1'''
Let $U \subset \mathbb R^n$. A [[Rectifiable varifold]] of dimension $m$ in $U$ is a couple $(M, \theta)$, where $M\subset U$ is an $m$-dimensional [[Rectifiable set]] and $\theta\colon M \to \mathbb R_+$ is a $\mathcal H^m$ measurable function, called density function. A varifold is called integral rectifiable if $\theta$ is integer valued.
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Let $U \subset \mathbb R^n$. A [[Rectifiable varifold]] $V$ of dimension $m$ in $U$ is a couple $(M, \theta)$, where $M\subset U$ is an $m$-dimensional [[Rectifiable set]] and $\theta\colon M \to \mathbb R_+$ is a $\mathcal H^m$ measurable function, called density function. A varifold is called integral rectifiable if $\theta$ is integer valued.
  
  
 
To a varifold we can naturally associate a measure defined and denoted by
 
To a varifold we can naturally associate a measure defined and denoted by
$$\mu_V(A):=\int_{M\cap A} \theta \,d\mathcal H^m\quad \text{for every Borel set } A\subset \mathbb R^n$$
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$$\mu_V(A):=\int_{M\cap A} \theta \,d\mathcal H^m,\quad \text{for every Borel set } A\subset \mathbb R^n.$$
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The mass of the varifold is defined by $\bf{M}(V):=\mu_V(U)$.
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 +
Notice that by
  
 
==First Variation and Stationariety==
 
==First Variation and Stationariety==

Revision as of 10:08, 11 September 2012

2020 Mathematics Subject Classification: Primary: 49Q15 [MSN][ZBL]


Rectifiable varifolds are a generalization of rectifiable sets in the sense that they allow for a density function to be defined on the set. They are also strictly connected to rectifiable currents, in fact to such a current one can always associate a varifold by putting aside the orientation.

Definitions

Definition 1 Let $U \subset \mathbb R^n$. A Rectifiable varifold $V$ of dimension $m$ in $U$ is a couple $(M, \theta)$, where $M\subset U$ is an $m$-dimensional Rectifiable set and $\theta\colon M \to \mathbb R_+$ is a $\mathcal H^m$ measurable function, called density function. A varifold is called integral rectifiable if $\theta$ is integer valued.


To a varifold we can naturally associate a measure defined and denoted by $$\mu_V(A):=\int_{M\cap A} \theta \,d\mathcal H^m,\quad \text{for every Borel set } A\subset \mathbb R^n.$$ The mass of the varifold is defined by $\bf{M}(V):=\mu_V(U)$.

Notice that by

First Variation and Stationariety

Allard's Regularity Theorem

References

[Sim] Leon Simon, "Lectures on Geometric Measure Theory". Proceedings of the centre for Mathematical Analysis. Australian National University, Canberra, 1983. MR0756417Zbl 0546.49019
[FX] Lin Fanghua, Yang Xiaoping, "Geometric Measure Theory-An Introduction". Advanced Mathematics Vol.1. International Press, Boston, 2002. MR2030862Zbl 1074.49011
How to Cite This Entry:
Luca.Spolaor/sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Luca.Spolaor/sandbox&oldid=27895