| Title:
|
Generalized versions of Ilmanen lemma: Insertion of $ C^{1,\omega} $ or $ C^{1,\omega}_{{\rm loc}} $ functions (English) |
| Author:
|
Kryštof, Václav |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
59 |
| Issue:
|
2 |
| Year:
|
2018 |
| Pages:
|
223-231 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
Ilmanen lemma |
| Keyword:
|
$ C^{1,\omega} $ function |
| Keyword:
|
semiconvex function with general modulus |
| MSC:
|
26B25 |
| idZBL:
|
Zbl 06940865 |
| idMR:
|
MR3815687 |
| DOI:
|
10.14712/1213-7243.2015.245 |
| . |
| Date available:
|
2018-06-28T08:45:50Z |
| Last updated:
|
2020-07-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147254 |
| . |
| Reference:
|
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| Reference:
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