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\(Q\)-Valued Functions Revisited
Camillo De Lellis, University of Zurich, Switzerland, and Emanuele Nunzio Spadaro, University of Bonn, Germany
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Memoirs of the American Mathematical Society
2011; 79 pp; softcover
Volume: 211
ISBN-10: 0-8218-4914-X
ISBN-13: 978-0-8218-4914-9
List Price: US$66
Individual Members: US$39.60
Institutional Members: US$52.80
Order Code: MEMO/211/991
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In this memoir the authors revisit Almgren's theory of \(Q\)-valued functions, which are functions taking values in the space \(\mathcal{A}_Q(\mathbb{R}^{n})\) of unordered \(Q\)-tuples of points in \(\mathbb{R}^{n}\).

In particular, the authors:

  • give shorter versions of Almgren's proofs of the existence of \(\mathrm{Dir}\)-minimizing \(Q\)-valued functions, of their Hölder regularity, and of the dimension estimate of their singular set;
  • propose an alternative, intrinsic approach to these results, not relying on Almgren's biLipschitz embedding \(\xi: \mathcal{A}_Q(\mathbb{R}^{n})\to\mathbb{R}^{N(Q,n)}\);
  • improve upon the estimate of the singular set of planar \(\mathrm{D}\)-minimizing functions by showing that it consists of isolated points.

Table of Contents

  • Introduction
  • The elementary theory of \(Q\)-valued functions
  • Almgren's extrinsic theory
  • Regularity theory
  • Intrinsic theory
  • The improved estimate of the singular set in \(2\) dimensions
  • Bibliography
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