The Free Boundary of the Monotone Follower

This paper identifies the free boundary arising in the two- dimensional monotone follower, cheap control problem. It proves that if a region of inaction $\cal A$ is of locally finite perimeter (LFP), then $\cal A$ can be replaced by a new region of inaction $\tilde{\cal A}$ whose boundary is locally $C^1$ (up to sets of lower dimension). It then gives conditions under which the hypothesis (LFP) holds. Furthermore, under these conditions even higher regularity of the free boundary is obtained, namely $C^{2,\alpha}$ except perhaps at a single corner point.