# Generalization of First Variation of Area

The area of an $m$-rectifiable varifold in $n$-dimensional space can be expressed in terms of the surface divergence. More precisely, if $M$ is $m$-rectifiable, $\Omega$ is open, $\eta$ is a $C^1_c$ vector field, and $\Phi_\varepsilon(x)=x+\varepsilon\eta(x)$,

$$\left.\frac{d}{d\varepsilon}\mathscr{H}^m(\Phi_\varepsilon(M\cap\Omega))\right|_{\varepsilon=0}=\int_A \eta d \mathscr{H}^m$$

where $A=M\cap\Omega$.

In codimension one, the normal velocity of $M$ under this variation will be $\eta\cdot n$, where $n$ is a normal vector.

My question: is it possible to generalize this result to normal velocities in $L^2$? I do know that if A generalization like this holds, it will be necessary to assume that $M$ has weak mean curvature in $L^2$.

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