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$Q$-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: $42.00 Click above image for expanded view$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.

• 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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