Generalized versions of Ilmanen lemma: Insertion of $C^{1,\omega }$ or $C_{\mathrm{loc}}^{1,\omega }$ functions

Kryštof, Václav. "Generalized versions of Ilmanen lemma: Insertion of $ C^{1,\omega } $ or $ C^{1,\omega }_{{\rm loc}} $ functions." Commentationes Mathematicae Universitatis Carolinae 59.2 (2018): 223-231. <http://eudml.org/doc/294261>.

@article{Kryštof2018,
abstract = {We prove that for a normed linear space $ X $, if $ f_1\colon X\rightarrow \mathbb \{R\} $ is continuous and semiconvex with modulus $ \omega $, $ f_2\colon X\rightarrow \mathbb \{R\} $ is continuous and semiconcave with modulus $ \omega $ and $f_1\le f_2 $, then there exists $ f\in C^\{1,\omega \}(X) $ such that $ f_1\le f\le f_2 $. Using this result we prove a generalization of Ilmanen lemma (which deals with the case $ \omega (t)=t $) to the case of an arbitrary nontrivial modulus $ \omega $. This generalization (where a $ C^\{1,\omega \}_\{\{loc\}\} $ function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.},
author = {Kryštof, Václav},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Ilmanen lemma; $ C^\{1,\omega \} $ function; semiconvex function with general modulus},
language = {eng},
number = {2},
pages = {223-231},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Generalized versions of Ilmanen lemma: Insertion of $ C^\{1,\omega \} $ or $ C^\{1,\omega \}_\{\{\rm loc\}\} $ functions},
url = {http://eudml.org/doc/294261},
volume = {59},
year = {2018},
}

TY - JOUR
AU - Kryštof, Václav
TI - Generalized versions of Ilmanen lemma: Insertion of $ C^{1,\omega } $ or $ C^{1,\omega }_{{\rm loc}} $ functions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2018
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 59
IS - 2
SP - 223
EP - 231
AB - We prove that for a normed linear space $ X $, if $ f_1\colon X\rightarrow \mathbb {R} $ is continuous and semiconvex with modulus $ \omega $, $ f_2\colon X\rightarrow \mathbb {R} $ is continuous and semiconcave with modulus $ \omega $ and $f_1\le f_2 $, then there exists $ f\in C^{1,\omega }(X) $ such that $ f_1\le f\le f_2 $. Using this result we prove a generalization of Ilmanen lemma (which deals with the case $ \omega (t)=t $) to the case of an arbitrary nontrivial modulus $ \omega $. This generalization (where a $ C^{1,\omega }_{{loc}} $ function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.
LA - eng
KW - Ilmanen lemma; $ C^{1,\omega } $ function; semiconvex function with general modulus
UR - http://eudml.org/doc/294261
ER -