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Annali di Matematica Pura ed Applicata

, Volume 183, Issue 1, pp 79–95 | Cite as

Regular selections for multiple-valued functions

  • Camillo De LellisEmail author
  • Carlo Romano GrisantiEmail author
  • Paolo TilliEmail author
Article

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 of 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 regularity.

Some related problems are also discussed.

Keywords

Equivalence Class Related Problem Finite Group Abstract Setting Regular Selection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2003

Authors and Affiliations

  1. 1.Scuola Normale SuperiorePisaItaly
  2. 2.Dipartimento di Matematica Applicata “U. Dini”Univesità di PisaPisaItaly

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