Calculus of Variations and Geometric Measure Theory
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C. De Lellis - E. Spadaro

Q-Valued functions revisited

created by delellis on 29 Feb 2008
modified on 24 May 2017


Published Paper

Inserted: 29 feb 2008
Last Updated: 24 may 2017

Journal: Memoirs of the American Mathematical Society
Volume: 991
Year: 2011


In this note we revisit Almgren's theory of Q-valued functions, that are functions taking values in the space $\mathcal{A}_Q (*R*^n)$ of unordered Q-tuples of points in $*R*^n$. In particular: \begin{itemize} \item we give shorter versions of Almgren's proofs of the existence of Dir-minimizing Q-valued functions, of their Hölder regularity and of the dimension estimate of their singular set; \item we propose an alternative intrinsic approach to these results, not relying on Almgren's biLipschitz embedding of $\mathcal{A}_Q (*R*^n)$ in the euclidean space; \item we improve upon the estimate of the singular set of planar Dir-minimizing functions by showing that it consists of isolated points.

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