The minimum cut problem between two distinguished nodes in a undirected planar network with limited capacities on both nodes and edges is discussed. The traditional method reduces the node-edge-capacity problem to an only-edge-capacity problem. But this method does not maintain the planarity of a planar network, so the special qualities of planar networks can not be used. The traditional algorithm runs in time O(n²logn). In this paper, approaches that fully use the planarity of planar networks are presented. For an s-t network in which the source and the sink are on a same face, the minimum cut problem to the shortest path problem in a planar graph is reduced, thus an algorithm that runs in time O(n) is obtained. For a common planar network, another method that reduces the node-edge-capacity problem to an only-edge-capacity problem is presented. This reduction method does not destroy the planarity of a planar network, so that the problem can be solved in time O(nlogn).