Title | Regular selections for multiple-valued functions. |

Publication Type | Journal Article |

Year of Publication | 2004 |

Authors | De Lellis C., Grisanti C.R, Tilli P. |

Journal | Annali di Matematica Pura ed Applicata |

Volume | 183 |

Pagination | 79–95 |

Publisher | Ann. Mat. Pura Appl. |

Type of Article | Q-valued |

ISSN | 0373-3114 |

Keywords | differentiability, modulus of continuity |

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,\ensuremathα regularity. Some related problems are also discussed. |

Notes | Ann. Mat. Pura Appl. (4) 183 (2004), no. 1, 79–95. |

URL | http://www.springerlink.com/content/hny2gvjembvuth8p/ |

DOI | 10.1007/s10231-003-0081-5 |

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