I found the following definition in a book (S. Osher, R. Fedkiw, "Level Set Methods and Dynamic Implicit Surfaces", p. 140):
[the context is reconstruction of surfaces from unorganized point sets]
Let $S$ denote a set of points in $\mathbb{R}^3$, and define $d(x) = \mathrm{dist}(x, S)$ the shortest distance between $x$ and any of the points in $S$. Consider the following energy function:
$$ E(\Gamma) = \left[ \int_\Gamma d^p(x)\ ds \right]^\frac{1}{p} \quad, \qquad 1 \le p \le \infty $$ in which $\Gamma$ is a surface and $ds$ is the surface area.
The author now gives an expression for the gradient flow of this energy functional:
$$ \frac{d\Gamma}{dt} = -\left[ \int_\Gamma d^p(x)\ ds \right]^{\frac{1}{p} - 1} d^{p-1}(x) \left[ \nabla d(x) \cdot N + \frac{1}{p}d(x)\kappa \right] N $$
In this, $N$ is the surface normal and $\kappa$ the mean curvature. Also, $d(x) \kappa$ is called the surface tension.
Now, I realize that I may be in a bit over my head here, but I really would like to know how this expression was derived... unfortunately, the book is somewhat terse on the basics.
My take on this is that the underlying equation is
$$ \frac{d\Gamma}{dt} = -\nabla_{\!F}\ E(\Gamma)$$
in which $F$ is the space of deformations, and $\nabla_{\!F}\ E(\Gamma)$ is the gradient vector field of $E$. It seems the derivation above somehow applies the chain rule to this equation...
I have some more information, but I honestly don't know how much of this is standard stuff. I would be grateful if someone could give me some hints as to how to arrive at the result given.