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