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

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Cambridge University Press, 09.08.2012 - 454 Seiten
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.
 

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Inhalt

Borel and Radon measures
14
Radon measures and continuous functions
31
Differentiation of Radon measures
51
Two further applications of diflerentiation theory
64
Area formula
76
GaussGreen theorem
89
Rectifiable sets and blowups of Radon measures
96
Tangential diflerentiability and the area formula
106
Equilibrium shapes of liquids and sessile drops
229
Anisotropic surface energies
258
REGULARITY THEORY AND ANALYSIS
275
Excess and the height bound
290
The Lipschitz approximation theorem
303
The reverse Poincare inequality
320
Harmonic approximation and excess improvement
337
Iteration partial regularity and singular sets
345

SETS OF FINITE PERIIVIETER
117
The coarea formula and the approximation theorem
145
The Euclidean isoperimetric problem
157
Reduced boundary and De Giorgis structure theorem
167
Federers theorem and comparison sets
183
First and second variation of perimeter
195
Slicing boundaries of sets of finite perimeter
215
Higher regularity theorems
357
Analysis of singularities
362
MINIlVIIZING CLUSTERS
391
Existence of minimizing clusters
398
Regularity of minimizing clusters
431
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Über den Autor (2012)

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

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