  We consider the standard first passage percolation model in the rescaled
graph $\mathbb{Z}^d/n$ for $d\geq2$ and a domain $\Omega$ of boundary $\Gamma$
in $\mathbb{R}^d$. Let $\Gamma ^1$ and $\Gamma ^2$ be two disjoint open subsets
of $\Gamma$, representing the parts of $\Gamma$ through which some water can
enter and escape from $\Omega$. A law of large numbers for the maximal flow
from $\Gamma ^1$ to $\Gamma ^2$ in $\Omega$ is already known. In this paper we
investigate the asymptotic behavior of a maximal stream and a minimal cutset. A
maximal stream is a vector measure $\vec{\mu}_n^{\max}$ that describes how the
maximal amount of fluid can cross $\Omega$. Under conditions on the regularity
of the domain and on the law of the capacities of the edges, we prove that the
sequence $(\vec{\mu}_n^{\max})_{n\geq1}$ converges a.s. to the set of the
solutions of a continuous deterministic problem of maximal stream in an
anisotropic network. A minimal cutset can been seen as the boundary of a set
$E_n^{\min}$ that separates $\Gamma ^1$ from $\Gamma ^2$ in $\Omega$ and whose
random capacity is minimal. Under the same conditions, we prove that the
sequence $(E_n^{\min})_{n\geq1}$ converges toward the set of the solutions of a
continuous deterministic problem of minimal cutset. We deduce from this a
continuous deterministic max-flow min-cut theorem and a new proof of the law of
large numbers for the maximal flow. This proof is more natural than the
existing one, since it relies on the study of maximal streams and minimal
cutsets, which are the pertinent objects to look at.
