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Keywords:
Ilmanen lemma; $ C^{1,\omega} $ function; semiconvex function with general modulus
Ilmanen lemma; $ C^{1,\omega} $ function; semiconvex function with general modulus
Summary:
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}_{{loc}} $ function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.
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