Difference between revisions of "User:Luca.Spolaor/sandbox"
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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. | + | 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 | ||
| − | $$\mu_V(A):=\int_{M\cap A} \theta \,d\mathcal H^m\quad \text{for every Borel set } A\subset \mathbb R^n$$ | + | $$\mu_V(A):=\int_{M\cap A} \theta \,d\mathcal H^m,\quad \text{for every Borel set } A\subset \mathbb R^n.$$ |
| + | The mass of the varifold is defined by $\bf{M}(V):=\mu_V(U)$. | ||
| + | |||
| + | Notice that by | ||
==First Variation and Stationariety== | ==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 |
Luca.Spolaor/sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Luca.Spolaor/sandbox&oldid=27895