How does the inverse mean curvature flow start with minimal surface? I am unclear how the inverse mean curvature flow starts with a minimal surface. If there is some point $p$ with mean curvature $H(P)=0$, how should we treat this problem and let the flow start?
 A: It isn't clear exactly what sort of initial conditions you're requiring. 
The difficulty with minimal initial conditions is part of the reason why it was an amazing result when Huisken--Ilmanen constructed a "weak inverse mean curvature flow" which 
(1) Can start at a minimal surface (technically, for certain things to work nicely, it should be outer-minimizing) 
(2) Exists for all time in an asymptotically flat manifold.
AND
(3) Still satisfies Geroch monotonicity, i.e. the Hawking mass is monotone along the flow.
That one could find a "flow" which satisfies (1) and (2) while still keeping (3) is incredible. 
Their paper is very readable, although quite long, so I'd recommend that you take a look at it, rather than I try to explain the ideas here.

EDIT: I've added some more information below. Its not totally clear what your motivation for the question is; if you add more information perhaps I can answer your question better. 

It is sometimes possible to define a classical flow which starts at a minimal surface. For example, in the Riemannian Schwarzschild metric 
$$
g = \left(1+\frac{2m}{r}\right)^4 \delta,
$$
on $\mathbb{R}^3\setminus \{0\}$, there is a inverse mean curvauture flow $\Sigma_t = \{r(t)\}\times \mathbb{S}^2$ defined for $t>0$, where $\lim_{t\searrow 0} r(t) = m/2$. I'll leave it to you to compute the associated ODE (HINT: the easiest way is to use the fact that $|\Sigma_t|=e^t|\Sigma_0|$. 
Be very careful with what I mean here for $t=0$. In particular, the PDE is not satisfied for $t=0$, just $t>0$. 

Here are some results about the classical flow:
http://www.ams.org/mathscinet-getitem?mr=1064876
http://www.ams.org/mathscinet-getitem?mr=1082861
http://www.ams.org/mathscinet-getitem?mr=1753358
