eBook ISBN:  9781470402785 
Product Code:  MEMO/144/687.E 
List Price:  $53.00 
MAA Member Price:  $47.70 
AMS Member Price:  $31.80 
eBook ISBN:  9781470402785 
Product Code:  MEMO/144/687.E 
List Price:  $53.00 
MAA Member Price:  $47.70 
AMS Member Price:  $31.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 144; 2000; 132 ppMSC: Primary 49; Secondary 28; 42
Roughly speaking, a \(d\)dimensional subset of \(\mathbf R^n\) is minimizing if arbitrary deformations of it (in a suitable class) cannot decrease its \(d\)dimensional volume. For quasiminimizing sets, one allows the mass to decrease, but only in a controlled manner. To make this precise we follow Almgren's notion of “restricted sets” [2]. Graphs of Lipschitz mappings \(f\:\mathbf R^d \to \mathbf R^{nd}\) are always quasiminimizing, and Almgren showed that quasiminimizing sets are rectifiable. Here we establish uniform rectifiability properties of quasiminimizing sets, which provide a more quantitative sense in which these sets behave like Lipschitz graphs. (Almgren also established stronger smoothness properties under tighter quasiminimality conditions.)
Quasiminimizing sets can arise as minima of functionals with highly irregular “coefficients”. For such functionals, one cannot hope in general to have much more in the way of smoothness or structure than uniform rectifiability, for reasons of bilipschitz invariance. (See also [9].)
One motivation for considering minimizers of functionals with irregular coefficients comes from the following type of question. Suppose that one is given a compact set \(K\) with upper bounds on its \(d\)dimensional Hausdorff measure, and lower bounds on its \(d\)dimensional topology. What can one say about the structure of \(K\)? To what extent does it behave like a nice \(d\)dimensional surface? A basic strategy for dealing with this issue is to first replace \(K\) by a set which is minimizing for a measurement of volume that imposes a large penalty on points which lie outside of \(K\). This leads to a kind of regularization of \(K\), in which cusps and very scattered parts of \(K\) are removed, but without adding more than a small amount from the complement of \(K\). The results for quasiminimizing sets then lead to uniform rectifiability properties of this regularization of \(K\).
To actually produce minimizers of general functionals it is sometimes convenient to work with (finite) discrete models. A nice feature of uniform rectifiability is that it provides a way to have bounds that cooperate robustly with discrete approximations, and which survive in the limit as the discretization becomes finer and finer.
ReadershipGraduate students and research mathematicians interested in calculus of variations and optimal control; optimization.

Table of Contents

Chapters

0. Introduction

1. Quasiminimizers

2. Uniform rectifiability and the main result

3. Lipschitz projections into skeleta

4. Local Ahlforsregularity

5. Lipschitz mappings with big images

6. From Lipschitz functions to projections

7. Regular sets and cubical patchworks

8. A stoppingtime argument

9. Proof of main Lemma 8.7

10. Big projections

11. Restricted and dyadic quasiminimizers

12. Applications


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Roughly speaking, a \(d\)dimensional subset of \(\mathbf R^n\) is minimizing if arbitrary deformations of it (in a suitable class) cannot decrease its \(d\)dimensional volume. For quasiminimizing sets, one allows the mass to decrease, but only in a controlled manner. To make this precise we follow Almgren's notion of “restricted sets” [2]. Graphs of Lipschitz mappings \(f\:\mathbf R^d \to \mathbf R^{nd}\) are always quasiminimizing, and Almgren showed that quasiminimizing sets are rectifiable. Here we establish uniform rectifiability properties of quasiminimizing sets, which provide a more quantitative sense in which these sets behave like Lipschitz graphs. (Almgren also established stronger smoothness properties under tighter quasiminimality conditions.)
Quasiminimizing sets can arise as minima of functionals with highly irregular “coefficients”. For such functionals, one cannot hope in general to have much more in the way of smoothness or structure than uniform rectifiability, for reasons of bilipschitz invariance. (See also [9].)
One motivation for considering minimizers of functionals with irregular coefficients comes from the following type of question. Suppose that one is given a compact set \(K\) with upper bounds on its \(d\)dimensional Hausdorff measure, and lower bounds on its \(d\)dimensional topology. What can one say about the structure of \(K\)? To what extent does it behave like a nice \(d\)dimensional surface? A basic strategy for dealing with this issue is to first replace \(K\) by a set which is minimizing for a measurement of volume that imposes a large penalty on points which lie outside of \(K\). This leads to a kind of regularization of \(K\), in which cusps and very scattered parts of \(K\) are removed, but without adding more than a small amount from the complement of \(K\). The results for quasiminimizing sets then lead to uniform rectifiability properties of this regularization of \(K\).
To actually produce minimizers of general functionals it is sometimes convenient to work with (finite) discrete models. A nice feature of uniform rectifiability is that it provides a way to have bounds that cooperate robustly with discrete approximations, and which survive in the limit as the discretization becomes finer and finer.
Graduate students and research mathematicians interested in calculus of variations and optimal control; optimization.

Chapters

0. Introduction

1. Quasiminimizers

2. Uniform rectifiability and the main result

3. Lipschitz projections into skeleta

4. Local Ahlforsregularity

5. Lipschitz mappings with big images

6. From Lipschitz functions to projections

7. Regular sets and cubical patchworks

8. A stoppingtime argument

9. Proof of main Lemma 8.7

10. Big projections

11. Restricted and dyadic quasiminimizers

12. Applications