Q-valued functions revisited

Details Q-valued functions revisited Q-valued functions revisited

2008-03-03

arxiv.org/abs/0803.0060v4

Memoirs of the AMS 211 (2011), no. 991

Mathematics

Analysis of PDEs

84 pages, 3 figures; changes: few typos corrected, reformatted for submission

Scientific paper

10.1090/S0065-9266-10-00607-1

In this note we revisit Almgren's theory of Q-valued functions, that are functions taking values in the space of unordered Q-tuples of points in R^n. In particular: 1) we give shorter versions of Almgren's proofs of the existence of Dir-minimizing Q-valued functions, of their Hoelder regularity and of the dimension estimate of their singular set; 2) we propose an alternative intrinsic approach to these results, not relying on Almgren's biLipschitz embedding; 3) 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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