Calculus of Variations and Geometric Measure Theory

C. De Lellis - C. R. Grisanti - P. Tilli

Regular selection for multiple valued functions

created on 20 Nov 2000
modified by delellis on 06 May 2011


Published Paper

Inserted: 20 nov 2000
Last Updated: 6 may 2011

Journal: Annali di Matematica Pura ed Applicata
Volume: 183
Pages: 79-95
Year: 2000


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 $\mathbf R^n$ 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 $C^{k,\alpha}$ regularity.

Some related problems are also discussed.

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