7.3 Reduced boundary and the Federer-Volpert theorem

The reduced boundary of a set is a crucial concept in geometric measure theory. It provides a more regular subset of the topological boundary, essential for studying geometric properties of sets. This concept is key to understanding sets of finite perimeter.

The Federer-Volpert theorem is a cornerstone result in this field. It states that the reduced boundary of a set with finite perimeter is countably rectifiable, bridging the gap between measure theory and geometry. This theorem has far-reaching implications for studying set boundaries.

Reduced Boundary of a Set

Definition and Properties

• The reduced boundary of a set E, denoted by ∂*E, is a subset of the topological boundary ∂E that satisfies certain regularity properties
• A point x belongs to the reduced boundary ∂*E if and only if:
  • The measure-theoretic outer unit normal νE(x) exists at x
    • νE(x) is defined as the limit of the unit vectors (y - x)/|y - x| as y approaches x from outside E
  • The Lebesgue density of E at x is equal to 1/2
    • Lebesgue density is the limit of the ratio of the measure of E intersected with a ball centered at x to the measure of the ball, as the radius tends to zero

Significance and Applications

• The reduced boundary provides a more regular and well-behaved subset of the topological boundary, essential for studying geometric and measure-theoretic properties of sets
• It is crucial in the formulation of:
  • Gauss-Green theorem for sets of finite perimeter
  • Divergence theorem for sets of finite perimeter
• Allows for the application of geometric measure theory results (area formula, coarea formula) to study properties of the reduced boundary
• Enables proving the existence of approximate tangent planes to ∂*E at Hn-1-almost every point

Federer-Volpert Theorem

Statement and Rectifiability

• The Federer-Volpert theorem states that if E is a set of finite perimeter in ℝn, then:
  • The reduced boundary ∂*E is countably (n-1)-rectifiable
    • A set is countably (n-1)-rectifiable if it can be covered, up to a set of Hn-1-measure zero, by a countable union of (n-1)-dimensional Lipschitz graphs
  • Hn-1(∂E \ ∂*E) = 0, where Hn-1 denotes the (n-1)-dimensional Hausdorff measure

Key Steps in the Proof

• Show that the measure-theoretic outer unit normal νE(x) exists Hn-1-almost everywhere on ∂*E
• Prove that the function x ↦ νE(x) is Borel measurable on ∂*E
• Use the Besicovitch differentiation theorem to show that the Lebesgue density of E is equal to 1/2 at Hn-1-almost every point of ∂*E
• Apply the De Giorgi structure theorem to conclude that:
  • ∂*E is countably (n-1)-rectifiable
  • Hn-1(∂E \ ∂*E) = 0

Regularity of Reduced Boundary

Implications of Federer-Volpert Theorem

• The Federer-Volpert theorem implies that the reduced boundary ∂*E is (n-1)-rectifiable, indicating a certain degree of regularity
• Rectifiability allows for the application of geometric measure theory results to study properties of ∂*E:
  • Area formula
  • Coarea formula
• The theorem can be used to prove the existence of approximate tangent planes to ∂*E at Hn-1-almost every point

Measurability of the Outer Unit Normal

• The Federer-Volpert theorem implies that the measure-theoretic outer unit normal νE(x) is:
  • Well-defined on ∂*E
  • Hn-1-measurable on ∂*E
• The measurability of νE(x) is essential for studying the behavior of the perimeter measure

Reduced vs Measure-Theoretic Boundary

Definitions and Relationship

• The measure-theoretic boundary, denoted by ∂ME, is the set of points where the Lebesgue density of E is neither 0 nor 1
• The reduced boundary ∂*E is a subset of the measure-theoretic boundary ∂ME
  • Lebesgue density of E is equal to 1/2 at every point of ∂*E
• The Federer-Volpert theorem implies that ∂*E and ∂ME coincide up to a set of Hn-1-measure zero

Generalization and Importance

• The measure-theoretic boundary ∂ME can be seen as a generalization of the reduced boundary ∂*E
  • Allows for a more general notion of the boundary of a set in terms of Lebesgue density
• The relationship between ∂*E and ∂ME is important for:
  • Understanding the structure of sets of finite perimeter
  • Studying the properties of the perimeter measure