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Sets of Finite Perimeter and Geometric Variational Problems
An Introduction to Geometric Measure Theory

Part of Cambridge Studies in Advanced Mathematics

  • Date Published: August 2012
  • availability: Available
  • format: Hardback
  • isbn: 9781107021037


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About the Authors
  • 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.

    • Easy access to a topic at the forefront of mathematical research
    • Takes the reader from the basic theory through to research-level topics
    • Expository style includes fully detailed proofs, as well as heuristic motivation and remarks
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    Reviews & endorsements

    '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

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    Product details

    • Date Published: August 2012
    • format: Hardback
    • isbn: 9781107021037
    • length: 476 pages
    • dimensions: 229 x 160 x 28 mm
    • weight: 0.79kg
    • contains: 90 b/w illus.
    • availability: Available
  • Table of Contents

    Part I. Radon Measures on Rn:
    1. Outer measures
    2. Borel and Radon measures
    3. Hausdorff measures
    4. Radon measures and continuous functions
    5. Differentiation of Radon measures
    6. Two further applications of differentiation theory
    7. Lipschitz functions
    8. Area formula
    9. Gauss–Green theorem
    10. Rectifiable sets and blow-ups of Radon measures
    11. Tangential differentiability and the area formula
    Part II. Sets of Finite Perimeter:
    12. Sets of finite perimeter and the Direct Method
    13. The coarea formula and the approximation theorem
    14. The Euclidean isoperimetric problem
    15. Reduced boundary and De Giorgi's structure theorem
    16. Federer's theorem and comparison sets
    17. First and second variation of perimeter
    18. Slicing boundaries of sets of finite perimeter
    19. Equilibrium shapes of liquids and sessile drops
    20. Anisotropic surface energies
    Part III. Regularity Theory and Analysis of Singularities:
    21. (Λ, r0)-perimeter minimizers
    22. Excess and the height bound
    23. The Lipschitz approximation theorem
    24. The reverse Poincaré inequality
    25. Harmonic approximation and excess improvement
    26. Iteration, partial regularity, and singular sets
    27. Higher regularity theorems
    28. Analysis of singularities
    Part IV. Minimizing Clusters:
    29. Existence of minimizing clusters
    30. Regularity of minimizing clusters

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    Sets of Finite Perimeter and Geometric Variational Problems

    Francesco Maggi

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  • Author

    Francesco Maggi, University of Texas, Austin
    Francesco Maggi is an Associate Professor at the University of Texas, Austin, USA.

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