I am using the following definition of the Global Minimum Cut problem: Given a graph $G = (V,E)$, a Cut of $G$ is a partition of $V$ into two subsets $(A,B)$. A cut-edge of $C$ is an edge $(u,v) \in E$, with $u \in A$ and $v \in B$. A Global Minimum Cut of $G$ is a Cut which is the Minimum in some metric.
For my purposes this metric is always considered to be a weight/capacity. For undirected graphs it is well known that the Minimum s-t-Cut from among all s-t-Cuts in the graph is equivalent to the Global Minimum Cut.
I was wondering whether or not the same approach has any application for directed graphs. In this case, it seems to me like a Global Minimum Cut seeks to destroy a graph's weak connectivity, whereas the approach over s-t-Cuts will only ever guarantee the destruction of a graph's strong connectivity. Am I missing anything that would solve this problem?