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$Q$-valued functions revisited


About this Title

Camillo De Lellis, Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190 CH-8057 Zürich and Emanuele Nunzio Spadaro, Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190 CH-8057 Zürich

Publication: Memoirs of the American Mathematical Society
Publication Year 2011: Volume 211, Number 991
ISBNs: 978-0-8218-4914-9 (print); 978-1-4704-0608-0 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-10-00607-1
Published electronically: July 27, 2010
MathSciNet review: 2663735
Keywords:$Q$-valued functions; Dirichlet energy; existence and regularity; metric spaces; harmonic maps
MSC (2000): Primary 49Q20, 35J55, 54E40, 53A10

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Table of Contents


Chapters

  • Introduction
  • Chapter 1. The elementary theory of $Q$-valued functions
  • Chapter 2. Almgren’s extrinsic theory
  • Chapter 3. Regularity theory
  • Chapter 4. Intrinsic theory
  • Chapter 5. The improved estimate of the singular set in 2 dimensions

Abstract


In this note we revisit Almgren’s theory of $Q$-valued functions, that 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: \begin{itemize} \item we 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; \item we propose an alternative, intrinsic approach to these results, not relying on Almgren’s biLipschitz embedding $\xi : \mathcal {A}_Q(\mathbb {R}^{n})\to \mathbb {R}^{N(Q,n)}$; \item we improve upon the estimate of the singular set of planar $\mathrm {D}$-minimizing functions by showing that it consists of isolated points. \end{itemize}

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