Geodesic
A note on propagation of singularities of semiconcave functions of two variables
Commentationes Mathematicae Universitatis Carolinae, Tome 51 (2010) no. 3, pp. 453-458.

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P. Albano and P. Cannarsa proved in 1999 that, under some applicable conditions, singularities of semiconcave functions in $\mathbb R^n$ 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.
Classification : 26B25, 35A21
Keywords: semiconcave functions; singularities 
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Zajíček, Luděk. A note on propagation of singularities of semiconcave functions of two variables. Commentationes Mathematicae Universitatis Carolinae, Tome 51 (2010) no. 3, pp. 453-458. http://geodesic.mathdoc.fr/item/CMUC_2010__51_3_a6/