# Regular selections for multiple-valued functions

De Lellis, C; Grisanti, C; Tilli, P (2004). Regular selections for multiple-valued functions. Annali di Matematica Pura ed Applicata, 183(1):79-95.

## Abstract

Given a multiple-valued function f, we deal with the problem of selecting its single valued branches. This problem can be stated in a rather abstract setting considering a metric space E and a finite group G of isometries of E. Given a function f which takes values in the equivalence classes of E/G, the problem consists in finding a map g with the same domain as f and taking values in E, such that at every point t the equivalence class of g(t) coincides with f(t). If the domain of f is an interval, we show the existence of a function g with these properties which, moreover, has the same modulus of continuity of f. In the particular case where E is the product of Q copies of ℝ n and G is the group of permutations of Q elements, it is possible to introduce a notion of differentiability for multiple valued functions. In this case, we prove that the function g can be constructed in such a way to preserve C k,α regularity. Some related problems are also discussed.

## Abstract

Given a multiple-valued function f, we deal with the problem of selecting its single valued branches. This problem can be stated in a rather abstract setting considering a metric space E and a finite group G of isometries of E. Given a function f which takes values in the equivalence classes of E/G, the problem consists in finding a map g with the same domain as f and taking values in E, such that at every point t the equivalence class of g(t) coincides with f(t). If the domain of f is an interval, we show the existence of a function g with these properties which, moreover, has the same modulus of continuity of f. In the particular case where E is the product of Q copies of ℝ n and G is the group of permutations of Q elements, it is possible to introduce a notion of differentiability for multiple valued functions. In this case, we prove that the function g can be constructed in such a way to preserve C k,α regularity. Some related problems are also discussed.

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## Additional indexing

Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Physical Sciences > Applied Mathematics modulus of continuity, differentiability English 2004 17 Sep 2010 07:26 29 Jul 2020 19:40 Springer 0373-3114 The original publication is available at www.springerlink.com Green https://doi.org/10.1007/s10231-003-0081-5 http://www.ams.org/mathscinet-getitem?mr=2044333

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