Rectifiability

Rectifiability
Title Rectifiability PDF eBook
Author Pertti Mattila
Publisher Cambridge University Press
Pages 182
Release 2023-01-12
Genre Mathematics
ISBN 1009288091

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Rectifiable sets, measures, currents and varifolds are foundational concepts in geometric measure theory. The last four decades have seen the emergence of a wealth of connections between rectifiability and other areas of analysis and geometry, including deep links with the calculus of variations and complex and harmonic analysis. This short book provides an easily digestible overview of this wide and active field, including discussions of historical background, the basic theory in Euclidean and non-Euclidean settings, and the appearance of rectifiability in analysis and geometry. The author avoids complicated technical arguments and long proofs, instead giving the reader a flavour of each of the topics in turn while providing full references to the wider literature in an extensive bibliography. It is a perfect introduction to the area for researchers and graduate students, who will find much inspiration for their own research inside.

Analytic Capacity, Rectifiability, Menger Curvature and Cauchy Integral

Analytic Capacity, Rectifiability, Menger Curvature and Cauchy Integral
Title Analytic Capacity, Rectifiability, Menger Curvature and Cauchy Integral PDF eBook
Author Hervé Pajot
Publisher Springer Science & Business Media
Pages 140
Release 2002-11-26
Genre Mathematics
ISBN 9783540000013

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Based on a graduate course given by the author at Yale University this book deals with complex analysis (analytic capacity), geometric measure theory (rectifiable and uniformly rectifiable sets) and harmonic analysis (boundedness of singular integral operators on Ahlfors-regular sets). In particular, these notes contain a description of Peter Jones' geometric traveling salesman theorem, the proof of the equivalence between uniform rectifiability and boundedness of the Cauchy operator on Ahlfors-regular sets, the complete proofs of the Denjoy conjecture and the Vitushkin conjecture (for the latter, only the Ahlfors-regular case) and a discussion of X. Tolsa's solution of the Painlevé problem.

Uniform Rectifiability and Quasiminimizing Sets of Arbitrary Codimension

Uniform Rectifiability and Quasiminimizing Sets of Arbitrary Codimension
Title Uniform Rectifiability and Quasiminimizing Sets of Arbitrary Codimension PDF eBook
Author Guy David
Publisher American Mathematical Soc.
Pages 146
Release 2000
Genre Mathematics
ISBN 0821820486

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This book is intended for graduate students and research mathematicians interested in calculus of variations and optimal control; optimization.

Rectifiable Sets, Densities and Tangent Measures

Rectifiable Sets, Densities and Tangent Measures
Title Rectifiable Sets, Densities and Tangent Measures PDF eBook
Author Camillo De Lellis
Publisher European Mathematical Society
Pages 140
Release 2008
Genre Mathematics
ISBN 9783037190449

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The characterization of rectifiable sets through the existence of densities is a pearl of geometric measure theory. The difficult proof, due to Preiss, relies on many beautiful and deep ideas and novel techniques. Some of them have already proven useful in other contexts, whereas others have not yet been exploited. These notes give a simple and short presentation of the former and provide some perspective of the latter. This text emerged from a course on rectifiability given at the University of Zurich. It is addressed both to researchers and students; the only prerequisite is a solid knowledge in standard measure theory. The first four chapters give an introduction to rectifiable sets and measures in Euclidean spaces, covering classical topics such as the area formula, the theorem of Marstrand and the most elementary rectifiability criterions. The fifth chapter is dedicated to a subtle rectifiability criterion due to Marstrand and generalized by Mattila, and the last three focus on Preiss' result. The aim is to provide a self-contained reference for anyone interested in an overview of this fascinating topic.

Analysis of and on Uniformly Rectifiable Sets

Analysis of and on Uniformly Rectifiable Sets
Title Analysis of and on Uniformly Rectifiable Sets PDF eBook
Author Guy David
Publisher American Mathematical Soc.
Pages 370
Release 1993
Genre Mathematics
ISBN 0821815377

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* The only available reference on uniform rectifiabilityThe text covers the understanding of uniform rectifiability of a given set in terms of the approximate behaviour of the set at most locations and scales.

Rectifiable Measures, Square Functions Involving Densities, and the Cauchy Transform

Rectifiable Measures, Square Functions Involving Densities, and the Cauchy Transform
Title Rectifiable Measures, Square Functions Involving Densities, and the Cauchy Transform PDF eBook
Author Xavier Tolsa
Publisher American Mathematical Soc.
Pages 142
Release 2017-01-18
Genre Mathematics
ISBN 1470422522

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This monograph is devoted to the proof of two related results. The first one asserts that if is a Radon measure in satisfyingfor -a.e. , then is rectifiable. Since the converse implication is already known to hold, this yields the following characterization of rectifiable sets: a set with finite -dimensional Hausdorff measure is rectifiable if and only ifH^1x2EThe second result of the monograph deals with the relationship between the above square function in the complex plane and the Cauchy transform . Assuming that has linear growth, it is proved that is bounded in if and only iffor every square .

Singular Sets of Minimizers for the Mumford-Shah Functional

Singular Sets of Minimizers for the Mumford-Shah Functional
Title Singular Sets of Minimizers for the Mumford-Shah Functional PDF eBook
Author Guy David
Publisher Springer Science & Business Media
Pages 592
Release 2006-03-10
Genre Mathematics
ISBN 3764373024

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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.