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

Luděk Zajíček

Commentationes Mathematicae Universitatis Carolinae (2010)

  • Volume: 51, Issue: 3, page 453-458
  • ISSN: 0010-2628

Abstract

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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) ψ ( x ) = ( x , y 1 ( x ) - y 2 ( x ) ) , x ∈ [ 0 , α ] , where y 1 , y 2 are convex and Lipschitz on [ 0 , α ] . In other words: singularities propagate along arcs with finite turn.

How to cite

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Zajíček, Luděk. "A note on propagation of singularities of semiconcave functions of two variables." Commentationes Mathematicae Universitatis Carolinae 51.3 (2010): 453-458. <http://eudml.org/doc/38141>.

@article{Zajíček2010,
abstract = {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.},
author = {Zajíček, Luděk},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {semiconcave functions; singularities; semiconcave function; singularity},
language = {eng},
number = {3},
pages = {453-458},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A note on propagation of singularities of semiconcave functions of two variables},
url = {http://eudml.org/doc/38141},
volume = {51},
year = {2010},
}

TY - JOUR
AU - Zajíček, Luděk
TI - A note on propagation of singularities of semiconcave functions of two variables
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2010
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 51
IS - 3
SP - 453
EP - 458
AB - 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.
LA - eng
KW - semiconcave functions; singularities; semiconcave function; singularity
UR - http://eudml.org/doc/38141
ER -

References

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  8. Pavlica D., On the points of non-differentiability of convex functions, Comment. Math. Univ. Carolin. 45 (2004), 727–734. Zbl1100.26006MR2103086
  9. Spingarn J.E., 10.1090/S0002-9947-1981-0597868-8, Trans. Amer. Math. Soc. 264 (1981), 77–89. Zbl0465.26008MR0597868DOI10.1090/S0002-9947-1981-0597868-8
  10. Veselý L., Zajíček L., Delta-convex mappings between Banach spaces and applications, Dissertationes Math. (Rozprawy Mat.) 289 (1989). MR1016045
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