|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|
|Publisher||Ann. Mat. Pura Appl.|
|Type of Article||Q-valued|
|Keywords||differentiability, modulus of continuity|
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.
Ann. Mat. Pura Appl. (4) 183 (2004), no. 1, 79–95.