## Václav KryštofGeneralized versions of Ilmanen lemma: Insertion of $C^{1,\omega}$ or $C^{1,\omega}_{{\rm loc}}$ functions

Comment.Math.Univ.Carolin. 59,2 (2018) 223-231.

Abstract:We prove that for a normed linear space $X$, if $f_1\colon X\to\mathbb{R}$ is continuous and semiconvex with modulus $\omega$, $f_2\colon X\to\mathbb{R}$ is continuous and semiconcave with modulus $\omega$ and $f_1\leq f_2$, then there exists $f\in C^{1,\omega}(X)$ such that $f_1\leq f\leq f_2$. 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 $C^{1,\omega}_{{\rm loc}}$ function is inserted) gives a positive answer to a~problem formulated by A.\ Fathi and M.\ Zavidovique in 2010.

Keywords: Ilmanen lemma; $C^{1,\omega}$ function; semiconvex function with general modulus

DOI: DOI 10.14712/1213-7243.2015.245
AMS Subject Classification: 26B25

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