Regular selections for multiple-valued functions
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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,α regularity.
Some related problems are also discussed.
KeywordsEquivalence Class Related Problem Finite Group Abstract Setting Regular Selection
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