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Uniform Rectifiability And Quasiminimizing Sets Of Arbitrary Codimension

Author: Guy David
Publisher: American Mathematical Soc.
ISBN: 0821820486
Size: 36.92 MB
Format: PDF
Category : Mathematics
Languages : en
Pages : 132
View: 2455

Roughly speaking, a $d$-dimensional subset of $\mathbfR^n$ 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' [{\textbold 2}]. Graphs of Lipschitz mappings $f\:\mathbfR^d \to \mathbfR^{n-d}$ 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 [{\textbold 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.

Uniform Rectifiability And Quasiminimizing Sets Of Arbitrary Codimension

Author: Guy David
Publisher: American Mathematical Soc.
ISBN: 0821820486
Size: 22.18 MB
Format: PDF, ePub
Category : Mathematics
Languages : en
Pages : 132
View: 4606

Roughly speaking, a $d$-dimensional subset of $\mathbfR^n$ 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' [{\textbold 2}]. Graphs of Lipschitz mappings $f\:\mathbfR^d \to \mathbfR^{n-d}$ 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 [{\textbold 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.

Author: Charles Fefferman
Publisher: Princeton University Press
ISBN: 0691159416
Size: 61.87 MB
Format: PDF
Category : Mathematics
Languages : en
Pages : 480
View: 1629

Princeton University's Elias Stein was the first mathematician to see the profound interconnections that tie classical Fourier analysis to several complex variables and representation theory. His fundamental contributions include the Kunze-Stein phenomenon, the construction of new representations, the Stein interpolation theorem, the idea of a restriction theorem for the Fourier transform, and the theory of Hp Spaces in several variables. Through his great discoveries, through books that have set the highest standard for mathematical exposition, and through his influence on his many collaborators and students, Stein has changed mathematics. Drawing inspiration from Stein’s contributions to harmonic analysis and related topics, this volume gathers papers from internationally renowned mathematicians, many of whom have been Stein’s students. The book also includes expository papers on Stein’s work and its influence. The contributors are Jean Bourgain, Luis Caffarelli, Michael Christ, Guy David, Charles Fefferman, Alexandru D. Ionescu, David Jerison, Carlos Kenig, Sergiu Klainerman, Loredana Lanzani, Sanghyuk Lee, Lionel Levine, Akos Magyar, Detlef Müller, Camil Muscalu, Alexander Nagel, D. H. Phong, Malabika Pramanik, Andrew S. Raich, Fulvio Ricci, Keith M. Rogers, Andreas Seeger, Scott Sheffield, Luis Silvestre, Christopher D. Sogge, Jacob Sturm, Terence Tao, Christoph Thiele, Stephen Wainger, and Steven Zelditch.

King Of The Lags

Author: David Ward
Publisher:
ISBN: 9780285629097
Size: 75.71 MB
Format: PDF, ePub, Docs
Category :
Languages : en
Pages : 176
View: 1292

Singular Sets Of Minimizers For The Mumford Shah Functional

Author: Guy David
Publisher: Springer Science & Business Media
ISBN: 3764373024
Size: 51.82 MB
Format: PDF
Category : Mathematics
Languages : en
Pages : 581
View: 5136

The Mumford-Shah functional was introduced in the 1980s as a tool for automatic image segmentation, but its study gave rise to many interesting questions of analysis and geometric measure theory. The main object under scrutiny is a free boundary K where the minimizer may have jumps. The book presents an extensive description of the known regularity properties of the singular sets K, and the techniques to get them. It is largely self-contained, and should be accessible to graduate students in analysis. The core of the book is composed of regularity results that were proved in the last ten years and which are presented in a more detailed and unified way.

Rapport

Author:
Publisher:
ISBN:
Size: 56.29 MB
Format: PDF, Docs
Category : Mathematics
Languages : en
Pages :
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Analysis And Geometry Of Metric Measure Spaces

Author: Galia Devora Dafni
Publisher: American Mathematical Soc.
ISBN: 0821894188
Size: 77.80 MB
Format: PDF, ePub, Docs
Category : Mathematics
Languages : en
Pages : 220
View: 944

This book contains lecture notes from most of the courses presented at the 50th anniversary edition of the Seminaire de Mathematiques Superieure in Montreal. This 2011 summer school was devoted to the analysis and geometry of metric measure spaces, and featured much interplay between this subject and the emergent topic of optimal transportation. In recent decades, metric measure spaces have emerged as a fruitful source of mathematical questions in their own right, and as indispensable tools for addressing classical problems in geometry, topology, dynamical systems, and partial differential equations. The summer school was designed to lead young scientists to the research frontier concerning the analysis and geometry of metric measure spaces, by exposing them to a series of minicourses featuring leading researchers who highlighted both the state-of-the-art and some of the exciting challenges which remain. This volume attempts to capture the excitement of the summer school itself, presenting the reader with glimpses into this active area of research and its connections with other branches of contemporary mathematics.

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ISBN:
Size: 43.43 MB
Format: PDF, ePub, Mobi
Category :
Languages : en
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Catalogue

Author: American Mathematical Society
Publisher:
ISBN:
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Format: PDF
Category : Mathematics
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S Minaire Quations Aux D Riv Es Partielles

Author:
Publisher:
ISBN:
Size: 62.72 MB
Format: PDF, ePub
Category : Differential equations, Partial
Languages : fr
Pages :
View: 1422