$Q$-Valued Functions Revisited
- ISBN-10
- 082184914X
- ISBN-13
- 9780821849149
- Category
- Dirichlet principle
- Pages
- 79
- Language
- English
- Published
- 2011
- Publisher
- American Mathematical Soc.
- Authors
- Camillo De Lellis, Emanuele Nunzio Spadaro
Description
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 Holder 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.
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