A note on propagation of singularities of semiconcave functions of two variables

TL;DR

This paper demonstrates that in two dimensions, singularities of semiconcave functions propagate along very regular arcs with finite turn, specifically in a form involving convex Lipschitz functions.

Contribution

It proves that for two-variable semiconcave functions, the propagation arcs are convex, Lipschitz, and have finite turn, refining previous regularity results.

Findings

Propagation arcs are convex and Lipschitz.

Arcs can be expressed as a difference of convex Lipschitz functions.

Singularities propagate along arcs with finite turn.

Abstract

P. Albano and P. Cannarsa proved in 1999 that, under some applicable conditions, singularities of semiconcave functions in $\R^n$ propagate along Lipschitz arcs. Further regularity properties of these arcs were proved by P. Cannarsa and Y. Yu in 2009. We prove that, for $n=2$, these arcs are very regular: they can be found in the form (in a suitable Cartesian coordinate system) $\psi(x) = (x, y_1(x)-y_2(x)), x \in [0,\alpha]$, where $y_1$, $y_2$ are convex and Lipschitz on $[0,\alpha]$. In other words: singularities propagate along arcs with finite turn.