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A note on propagation of singularities of semiconcave functions of two variables. (English) Zbl 1224.26047

A function \(u\: \mathbb R^n\to \mathbb R\) is called semiconcave if it can be locally expressed in the form \(u(x)=g(x)+K\| x\| ^2\), where \(g\) is concave. For such functions, P. Albano and P. Cannarsa [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 28, No. 4, 719–740 (1999; Zbl 0957.26002)] proved that point singularities of a certain type continue along Lipschitz arcs. In the current paper, the author establishes further regularity of such arcs, namely, if the space dimension is two, under a suitable choice of coordinate system, they are locally represented as a difference of two convex functions.
Reviewer: Jan Malý (Praha)

MSC:

26B25 Convexity of real functions of several variables, generalizations
35A21 Singularity in context of PDEs

Citations:

Zbl 0957.26002
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