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

الأصل · EN

P. Albano and P. Cannarsa proved in 1999 that, under some applicable conditions, singularities of semiconcave functions in 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) ψ(x) = (x, y₁(x)-y₂(x)), x ∈ [0,α], where y₁, y₂ are convex and Lipschitz on [0,α]. In other words: singularities propagate along arcs with finite turn.