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$Q$-Valued Functions Revisited
 
Camillo De Lellis University of Zurich, Zurich, Switzerland
Emanuele Nunzio Spadaro University of Bonn, Bonn, Germany
Front Cover for $Q$-Valued Functions Revisited
Available Formats:
Electronic ISBN: 978-1-4704-0608-0
Product Code: MEMO/211/991.E
79 pp 
List Price: $70.00
MAA Member Price: $63.00
AMS Member Price: $42.00
Front Cover for $Q$-Valued Functions Revisited
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  • Front Cover for $Q$-Valued Functions Revisited
  • Back Cover for $Q$-Valued Functions Revisited
$Q$-Valued Functions Revisited
Camillo De Lellis University of Zurich, Zurich, Switzerland
Emanuele Nunzio Spadaro University of Bonn, Bonn, Germany
Available Formats:
Electronic ISBN:  978-1-4704-0608-0
Product Code:  MEMO/211/991.E
79 pp 
List Price: $70.00
MAA Member Price: $63.00
AMS Member Price: $42.00
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2112011
    MSC: Primary 49; 35; 54; 53;



    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
     
     
    • Chapters
    • Introduction
    • 1. The elementary theory of $Q$-valued functions
    • 2. Almgren’s extrinsic theory
    • 3. Regularity theory
    • 4. Intrinsic theory
    • 5. The improved estimate of the singular set in $2$ dimensions
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Volume: 2112011
MSC: Primary 49; 35; 54; 53;



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.

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