Q-valued functions revisited

2008/03/31 by Camillo De Lellis, Emanuele Nunzio Spadaro · 3 citations
Mathematics · #math.AP #msc:49Q20 #msc:35J55 #msc:54E40 #msc:53A10

paper · pdf · doi:10.1090/s0065-9266-10-00607-1

published as Memoirs of the AMS 211 (2011), no. 991 · 84 pages, 3 figures; changes: few typos corrected, reformatted for submission

arxiv created 2011/03/17 · arxiv updated 2011/03/18

Abstract

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 Rⁿ. 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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• Quantified compactness in Lipschitz-free spaces of [-1,1]ⁿ
• Hausdorff measure bounds for density-Q flat singularities of minimizing integral currents
• Solutions to the Thin Obstacle Problem with non-2D frequency

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