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

Book Description

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.

Table of Contents

  1. Cover Page
  2. Title Page
  3. Copyright Page
  4. Dedication Page
  5. Contents
  6. Preface
  7. Notation
  8. Part I Radon Measures on Rn
    1. 1 - Outer measures
      1. 1.1 - Examples of outer measures
      2. 1.2 - Measurable sets and σ-additivity
      3. 1.3 - Measure Theory and integration
    2. 2 - Borel and Radon measures
      1. 2.1 - Borel measures and Carathéodory's criterion
      2. 2.2 - Borel regular measures
      3. 2.3 - Approximation theorems for Borel measures
      4. 2.4 - Radon measures. Restriction, support, and push-forward
    3. 3 - Hausdorff measures
      1. 3.1 - Hausdorff measures and the notion of dimension
      2. 3.2 - H1 and the classical notion of length
      3. 3.3 - Hn = Ln and the isodiametric inequality
    4. 4 - Radon measures and continuous functions
      1. 4.1 - Lusin's theorem and density of continuous functions
      2. 4.2 - Riesz's theorem and vector-valued Radon measures
      3. 4.3 - Weak-star convergence
      4. 4.4 - Weak-star compactness criteria
      5. 4.5 - Regularization of Radon measures
    5. 5 - Differentiation of Radon measures
      1. 5.1 - Besicovitch's covering theorem
      2. 5.2 - Lebesgue-Besicovitch differentiation theorem
      3. 5.3 - Lebesgue points
    6. 6 - Two further applications of differentiation theory
      1. 6.1 - Campanato's criterion
      2. 6.2 - Lower dimensional densities of a Radon measure
    7. 7 - Lipschitz functions
      1. 7.1 - Kirszbraun's theorem
      2. 7.2 - Weak gradients
      3. 7.3 - Rademacher's theorem
    8. 8 - Area formula
      1. 8.1 - Area formula for linear functions
      2. 8.2 - The role of the singular set J f = 0
      3. 8.3 - Linearization of Lipschitz immersions
      4. 8.4 - Proof of the area formula
      5. 8.5 - Area formula with multiplicities
    9. 9 - Gauss-Green theorem
      1. 9.1 - Area of a graph of codimension one
      2. 9.2 - Gauss-Green theorem on open sets with C1-boundary
      3. 9.3 - Gauss-Green theorem on open sets with almost C1-boundary
    10. 10 - Rectifiable sets and blow-ups of Radon measures
      1. 10.1 - Decomposing rectifiable sets by regular Lipschitz images
      2. 10.2 - Approximate tangent spaces to rectifiable sets
      3. 10.3 - Blow-ups of Radon measures and rectifiability
    11. 11 - Tangential differentiability and the area formula
      1. 11.1 - Area formula on surfaces
      2. 11.2 - Area formula on rectifiable sets
      3. 11.3 - Gauss-Green theorem on surfaces
      4. Notes
  9. Part II Sets of Finite Perimeter
    1. 12 - Sets of finite perimeter and the Direct Method
      1. 12.1 - Lower semicontinuity of perimeter
      2. 12.2 - Topological boundary and Gauss-Green measure
      3. 12.3 - Regularization and basic set operations
      4. 12.4 - Compactness from perimeter bounds
      5. 12.5 - Existence of minimizers in geometric variational problems
      6. 12.6 - Perimeter bounds on volume
    2. 13 - The coarea formula and the approximation theorem
      1. 13.1 - The coarea formula
      2. 13.2 - Approximation by open sets with smooth boundary
      3. 13.3 - The Morse-Sard lemma
    3. 14 - The Euclidean isoperimetric problem
      1. 14.1 - Steiner inequality
      2. 14.2 - Proof of the Euclidean isoperimetric inequality
    4. 15 - Reduced boundary and De Giorgi's structure theorem
      1. 15.1 - Tangential properties of the reduced boundary
      2. 15.2 - Structure of Gauss-Green measures
    5. 16 - Federer's theorem and comparison sets
      1. 16.1 - Gauss-Green measures and set operations
      2. 16.2 - Density estimates for perimeter minimizers
    6. 17 - First and second variation of perimeter
      1. 17.1 - Sets of finite perimeter and diffeomorphisms
      2. 17.2 - Taylor's expansion of the determinant close to the identity
      3. 17.3 - First variation of perimeter and mean curvature
      4. 17.4 - Stationary sets and monotonicity of density ratios
      5. 17.5 - Volume-constrained perimeter minimizers
      6. 17.6 - Second variation of perimeter
    7. 18 - Slicing boundaries of sets of finite perimeter
      1. 18.1 - The coarea formula revised
      2. 18.2 - The coarea formula on Hn-1-rectifiable sets
      3. 18.3 - Slicing perimeters by hyperplanes
    8. 19 - Equilibrium shapes of liquids and sessile drops
      1. 19.1 - Existence of minimizers and Young's law
      2. 19.2 - The Schwartz inequality
      3. 19.3 - A constrained relative isoperimetric problem
      4. 19.4 - Liquid drops in the absence of gravity
      5. 19.5 - A symmetrization principle
      6. 19.6 - Sessile liquid drops
    9. 20 - Anisotropic surface energies
      1. 20.1 - Basic properties of anisotropic surface energies
      2. 20.2 - The Wulff problem
      3. 20.3 - Reshetnyak's theorems
      4. Notes
  10. Part III Regularity Theory and Analysis of Singularities
    1. 21 - (Λ, r0)-perimeter minimizers
      1. 21.1 - Examples of (Λ, r0)-perimeter minimizers
      2. 21.2 - (Λ, r0) and local perimeter minimality
      3. 21.3 - The C1,γ-reguarity theorem
      4. 21.4 - Density estimates for (Λ, r0)-perimeter minimizers
      5. 21.5 - Compactness for sequences of (Λ, r0)-perimeter minimizers
    2. 22 - Excess and the height bound
      1. 22.1 - Basic properties of the excess
      2. 22.2 - The height bound
    3. 23 - The Lipschitz approximation theorem
      1. 23.1 - The Lipschitz graph criterion
      2. 23.2 - The area functional and the minimal surfaces equation
      3. 23.3 - The Lipschitz approximation theorem
    4. 24 - The reverse Poincaré inequality
      1. 24.1 - Construction of comparison sets, part one
      2. 24.2 - Construction of comparison sets, part two
      3. 24.3 - Weak reverse Poincare inequality
      4. 24.4 - Proof of the reverse Poincare inequality
    5. 25 - Harmonic approximation and excess improvement
      1. 25.1 - Two lemmas on harmonic functions
      2. 25.2 - The “excess improvement by tilting” estimate
    6. 26 - Iteration, partial regularity, and singular sets
      1. 26.1 - The C1,γ-regularity theorem in the case Λ = 0
      2. 26.2 - The C1,γ-regularity theorem in the case Λ > 0
      3. 26.3 - C1,γ-regularity of the reduced boundary, and the characterization of the singular set
      4. 26.4 - C1-convergence for sequences of (Λ, r0)-perimeter minimizers
    7. 27 - Higher regularity theorems
      1. 27.1 - Elliptic equations for derivatives of Lipschitz minimizers
      2. 27.2 - Some higher regularity theorems
    8. 28 - Analysis of singularities
      1. 28.1 - Existence of densities at singular points
      2. 28.2 - Blow-ups at singularities and tangent minimal cones
      3. 28.3 - Simons' theorem
      4. 28.4 - Federer's dimension reduction argument
      5. 28.5 - Dimensional estimates for singular sets
      6. 28.6 - Examples of singular minimizing cones
      7. 28.7 - A Bernstein-type theorem
      8. Notes
  11. Part IV Minimizing Clusters
    1. 29 - Existence of minimizing clusters
      1. 29.1 - Definitions and basic remarks
      2. 29.2 - Strategy of proof
      3. 29.3 - Nucleation lemma
      4. 29.4 - Truncation lemma
      5. 29.5 - Infinitesimal volume exchanges
      6. 29.6 - Volume-fixing variations
      7. 29.7 - Proof of the existence of minimizing clusters
    2. 30 - Regularity of minimizing clusters
      1. 30.1 - Infiltration lemma
      2. 30.2 - Density estimates
      3. 30.3 - Regularity of planar clusters
      4. Notes
  12. References
  13. Index