**Memoirs of the American Mathematical Society**

2011;
79 pp;
Softcover

MSC: Primary 49; 35; 54; 53;

Print ISBN: 978-0-8218-4914-9

Product Code: MEMO/211/991

List Price: $70.00

AMS Member Price: $42.00

MAA Member Price: $63.00

**Electronic ISBN: 978-1-4704-0608-0
Product Code: MEMO/211/991.E**

List Price: $70.00

AMS Member Price: $42.00

MAA Member Price: $63.00

# \(Q\)-Valued Functions Revisited

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*Camillo De Lellis; Emanuele Nunzio Spadaro*

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.