Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory: 135 (Cambridge Studies in Advanced Mathematics, Series Number 135) - Hardcover

Maggi, Francesco

Hardcover

ISBN 10: 1107021030 ISBN 13: 9781107021037

Publisher: Cambridge University Press, 2012

Synopsis

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.

About this title

Review

'The book is a clear exposition of the theory of sets of finite perimeter, that introduces this topic in a very elegant and original way, and shows some deep and important results and applications ... Although most of the results contained in this book are classical, some of them appear in this volume for the first time in book form, and even the more classical topics which one may find in several other books are presented here with a strong touch of originality which makes this book pretty unique ... I strongly recommend this excellent book to every researcher or graduate student in the field of calculus of variations and geometric measure theory.' Alessio Figalli, Canadian Mathematical Society Notes

'The first aim of the book is to provide an introduction for beginners to the theory of sets of finite perimeter, presenting results concerning the existence, symmetry, regularity and structure of singularities in some variational problems involving length and area ... The secondary aim ... is to provide a multi-leveled introduction to the study of other variational problems ... an interested reader is able to enter with relative ease several parts of geometric measure theory and to apply some tools from this theory in the study of other problems from mathematics ... This is a well-written book by a specialist in the field ... It provides generous guidance to the reader [and] is recommended ... not only to beginners who can find an up-to-date source in the field but also to specialists ... It is an invitation to understand and to approach some deep and difficult problems from mathematics and physics.' Vasile Oproiu, Zentralblatt MATH

Book Description

This engaging graduate-level introduction to geometric measure theory bridges analysis and geometry, taking readers from basic theory to some of the most celebrated results in modern analysis. Explanatory pictures, detailed proofs, exercises and helpful remarks make it suitable for self-study and also a useful reference for researchers.

Edition details

• Publisher: Cambridge University Press
• Publication date: 2012
• ISBN 10: 1107021030
• ISBN 13: 9781107021037
• Binding: Hardcover
• Language: English
• Number of pages: 476