View source for User:Luca.Spolaor/sandbox

{{MSC|49Q15}}

[[Category:Classical measure theory]]

{{TEX|done}}
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 ${\bf M}(V):=\mu_V(U)$. 
The support of a varifold is defined by ${\rm spt}(V):={\rm spt}(\mu_V)$.

We can also define almost everywhere the tangent plane to a varifold by setting $T_xV:=T_xM$ at each point $x\in {\rm spt}(V)$ where the tangent plane to $M$ exists (see [[Rectifiable set]], Proposition 7).

The following are standard construction of Geometric Measure Theory:
* Restriction of a varifold to a set $A \subset \mathbb \R^n$ defined by
$$V A:=(A\cap M,\theta|_A)$$
* The push-forward of a varifold via a smooth map defined by
$$f^\#(V):=(f(M), \tilde{\theta}), \quad \text{where }\tilde{\theta}$$

==First Variation and Stationariety==

Consider the vector field $X\in C^1_c(U; \mathbb R^n)$, then we can define a one parameter family of diffeomorphism as the unique solution of

==Allard's Regularity Theorem==

==References==

{|
|-
|valign="top"|{{Ref|Sim}}||    Leon Simon, "Lectures on Geometric Measure Theory". Proceedings of the centre for Mathematical Analysis. Australian National University, Canberra, 1983.      {{MR|0756417}}{{ZBL|0546.49019}} 
|-
|valign="top"|{{Ref|FX}}||    Lin Fanghua, Yang Xiaoping, "Geometric Measure Theory-An Introduction". Advanced Mathematics Vol.1. International Press, Boston, 2002.      {{MR|2030862}}{{ZBL|1074.49011}} 
|}