The Free Boundary of the Monotone Follower

Abstract

This paper identifies the free boundary arising in the two-dimensional monotone follower, cheap control problem It proves that if a region of inaction $\mathcal{A}$ is of locally finite perimeter (LFP), then $\mathcal{A}$ can be replaced by a new region of inaction $\tilde {\mathcal{A}}$ 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.