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

An engaging graduate-level introduction that bridges analysis and geometry. Suitable for self-study and a useful reference for researchers.

Francesco Maggi (Author)

9781107021037, Cambridge University Press

Hardback, published 9 August 2012

476 pages, 90 b/w illus.
22.9 x 16 x 2.8 cm, 0.79 kg

'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

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.

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. (?, 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
References
Index.

Subject Areas: Euclidean geometry [PBMH], Geometry [PBM], Calculus of variations [PBKQ], Differential calculus & equations [PBKJ], Mathematics [PB]