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$Q$Valued Functions Revisited
eBook ISBN:  9781470406080 
Product Code:  MEMO/211/991.E 
List Price:  $70.00 
MAA Member Price:  $63.00 
AMS Member Price:  $42.00 
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$Q$Valued Functions Revisited
eBook ISBN:  9781470406080 
Product Code:  MEMO/211/991.E 
List Price:  $70.00 
MAA Member Price:  $63.00 
AMS Member Price:  $42.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 211; 2011; 79 ppMSC: 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: 211; 2011; 79 pp
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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