Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure TheoryCambridge University Press, 09.08.2012 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. |
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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Sets of Finite Perimeter and Geometric Variational Problems: An Introduction ... Francesco Maggi Eingeschränkte Leseprobe - 2012 |
Häufige Begriffe und Wortgruppen
a.e. t E R apply area formula argument balls Borel measure Borel set bounded Chapter coarea compact set conclude convergence convex Corollary countable deduce define definition denote differentiable disjoint divergence theorem Euclidean isoperimetric example Exercise exists Figure finally find first fix Fubini’s theorem geometric Given half-space Hausdorff measures Hence holds true I P(E implies isoperimetric inequality isoperimetric problem k-dimensional Lebesgue measurable set Lebesgue point Lemma linear functional Lipschitz function locally finite perimeter lower semicontinuity Math mean curvature minimizing clusters Moreover N-cluster open set outer measure P(Eh particular perimeter minimizer positive constants proof of Theorem Proposition 12.15 prove r0)-perimeter minimizer Radon measure rectifiable sets reduced boundary relative isoperimetric Remark satisfies Section sequence set E C R set F set of finite set of locally singular minimizing cone singular set Step three suffices tangent variational problems vE(x vector field