RECTIFIABLE SETS, DENSITIES, AND TANGENT MEASURES. PDF Download

Author: CAMILLO DE LELLIS.
Publisher:
ISBN: 9783037195444
Category :
Languages : en
Pages :

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Author: Camillo De Lellis
Publisher: European Mathematical Society
ISBN: 9783037190449
Category : Mathematics
Languages : en
Pages : 140

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Book Description
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.

Author: Adrian Martin
Publisher:
ISBN:
Category :
Languages : en
Pages :

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A major theme in geometric measure theory is establishing global properties, such as rectifiability, of sets or measures from local ones, such as densities or tangent measures. In establishing sufficient conditions for rectifiability it is useful to know what local properties are possible in a given setting, and this is the theme of this thesis. It is known, for 1-dimensional subsets of the plane with positive lower density, that the tangent measures being concentrated on a line is sufficient to imply rectifiability. It is shown here that this cannot be relaxed too much by demonstrating the existence of a 1-dimensional subset of the plane with positive lower density whose tangent measures are concentrated on the union of two halflines, and yet the set is unrectiable. A class of metrics are also defined on R, which are functions of the Euclidean metric, to give spaces of dimension s (s > 1), where the lower density is strictly greater than 21-s, and a method for gaining an explicit lower bound for a given dimension is developed. The results are related to the generalised Besicovitch 1/2 conjecture. Set functions are defined that measure how easily the subsets of a set can be covered by balls (of any radius) with centres in the subset. These set functions are studied and used to give lower bounds on the upper density of subsets of a normed space, in particular Euclidean spaces. Further attention is paid to subsets of R, where more explicit bounds are given.

Author: Pertti Mattila
Publisher: Cambridge University Press
ISBN: 9780521655958
Category : Mathematics
Languages : en
Pages : 360

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This book studies the geometric properties of general sets and measures in euclidean space.

Author: Xavier Tolsa
Publisher: American Mathematical Soc.
ISBN: 1470422522
Category : Cauchy transform
Languages : en
Pages : 130

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Book Description
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 .

Author: Pertti Mattila
Publisher: Cambridge University Press
ISBN: 1316583694
Category : Mathematics
Languages : en
Pages : 358

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Now in paperback, the main theme of this book is the study of geometric properties of general sets and measures in euclidean spaces. Applications of this theory include fractal-type objects such as strange attractors for dynamical systems and those fractals used as models in the sciences. The author provides a firm and unified foundation and develops all the necessary main tools, such as covering theorems, Hausdorff measures and their relations to Riesz capacities and Fourier transforms. The last third of the book is devoted to the Beisovich-Federer theory of rectifiable sets, which form in a sense the largest class of subsets of euclidean space posessing many of the properties of smooth surfaces. These sets have wide application including the higher-dimensional calculus of variations. Their relations to complex analysis and singular integrals are also studied. Essentially self-contained, this book is suitable for graduate students and researchers in mathematics.

Author: Pertti Mattila
Publisher: Cambridge University Press
ISBN: 1009288091
Category : Mathematics
Languages : en
Pages : 182

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Book Description
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.

Author: Frank Morgan
Publisher: Academic Press
ISBN: 0128045272
Category : Mathematics
Languages : en
Pages : 274

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Book Description
Geometric Measure Theory: A Beginner's Guide, Fifth Edition provides the framework readers need to understand the structure of a crystal, a soap bubble cluster, or a universe. The book is essential to any student who wants to learn geometric measure theory, and will appeal to researchers and mathematicians working in the field. Brevity, clarity, and scope make this classic book an excellent introduction to more complex ideas from geometric measure theory and the calculus of variations for beginning graduate students and researchers. Morgan emphasizes geometry over proofs and technicalities, providing a fast and efficient insight into many aspects of the subject, with new coverage to this edition including topical coverage of the Log Convex Density Conjecture, a major new theorem at the center of an area of mathematics that has exploded since its appearance in Perelman's proof of the Poincaré conjecture, and new topical coverage of manifolds taking into account all recent research advances in theory and applications. Focuses on core geometry rather than proofs, paving the way to fast and efficient insight into an extremely complex topic in geometric structures Enables further study of more advanced topics and texts Demonstrates in the simplest possible way how to relate concepts of geometric analysis by way of algebraic or topological techniques Contains full topical coverage of The Log-Convex Density Conjecture Comprehensively updated throughout

Author: Xavier Tolsa
Publisher: Springer Science & Business Media
ISBN: 3319005960
Category : Mathematics
Languages : en
Pages : 402

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This book studies some of the groundbreaking advances that have been made regarding analytic capacity and its relationship to rectifiability in the decade 1995–2005. The Cauchy transform plays a fundamental role in this area and is accordingly one of the main subjects covered. Another important topic, which may be of independent interest for many analysts, is the so-called non-homogeneous Calderón-Zygmund theory, the development of which has been largely motivated by the problems arising in connection with analytic capacity. The Painlevé problem, which was first posed around 1900, consists in finding a description of the removable singularities for bounded analytic functions in metric and geometric terms. Analytic capacity is a key tool in the study of this problem. In the 1960s Vitushkin conjectured that the removable sets which have finite length coincide with those which are purely unrectifiable. Moreover, because of the applications to the theory of uniform rational approximation, he posed the question as to whether analytic capacity is semiadditive. This work presents full proofs of Vitushkin’s conjecture and of the semiadditivity of analytic capacity, both of which remained open problems until very recently. Other related questions are also discussed, such as the relationship between rectifiability and the existence of principal values for the Cauchy transforms and other singular integrals. The book is largely self-contained and should be accessible for graduate students in analysis, as well as a valuable resource for researchers.

Author: Francesco Maggi
Publisher: Cambridge University Press
ISBN: 1139560891
Category : Mathematics
Languages : en
Pages : 475

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Book Description
The marriage of analytic power to geometric intuition drives many of today's mathematical advances, yet books that build the connection from an elementary level remain scarce. This engaging introduction to geometric measure theory bridges analysis and geometry, taking readers from basic theory to some of the most celebrated results in modern analysis. The theory of sets of finite perimeter provides a simple and effective framework. Topics covered include existence, regularity, analysis of singularities, characterization and symmetry results for minimizers in geometric variational problems, starting from the basics about Hausdorff measures in Euclidean spaces and ending with complete proofs of the regularity of area-minimizing hypersurfaces up to singular sets of codimension 8. Explanatory pictures, detailed proofs, exercises and remarks providing heuristic motivation and summarizing difficult arguments make this graduate-level textbook suitable for self-study and also a useful reference for researchers. Readers require only undergraduate analysis and basic measure theory.