7.2 Caccioppoli sets and the structure theorem

Caccioppoli sets are measurable sets with finite perimeter, defined using distributional derivatives. They're crucial in geometric measure theory, providing a way to study sets with irregular boundaries and connect measure-theoretic and geometric concepts.

The structure theorem for Caccioppoli sets reveals that their reduced boundaries are rectifiable, meaning they can be covered by countably many smooth hypersurfaces. This theorem links the perimeter of a set to the Hausdorff measure of its boundary, bridging analysis and geometry.

Caccioppoli sets: Definition and Properties

Definition and Perimeter

• A Caccioppoli set (set of finite perimeter) is a measurable set $E \subset \mathbb{R}^n$ where the distributional derivative of its characteristic function $\mathbf{1}_E$ is a vector-valued Radon measure with finite total variation
  • The characteristic function $\mathbf{1}_E$ equals 1 inside the set $E$ and 0 outside
  • The distributional derivative captures the "boundary" of the set in a weak sense
• The perimeter of a Caccioppoli set $E$ in an open set $\Omega \subset \mathbb{R}^n$ is $Per(E, \Omega) = |D\mathbf{1}_E|(\Omega)$
  • $|D\mathbf{1}_E|$ denotes the total variation of the distributional derivative of $\mathbf{1}_E$
  • Perimeter measures the size of the boundary of $E$ within $\Omega$

Closure and Boundary Properties

• Caccioppoli sets are closed under local convergence in measure
  • If a sequence of Caccioppoli sets $(E_i)$ converges locally in measure to a set $E$, then $E$ is also a Caccioppoli set
  • This property allows approximating Caccioppoli sets with smooth sets
• The reduced boundary $\partial^*E$ of a Caccioppoli set $E$ is a subset of the topological boundary $\partial E$
  • $\partial^*E$ has finite $(n-1)$-dimensional Hausdorff measure $\mathcal{H}^{n-1}$
  • The reduced boundary captures the "essential" part of the boundary
• The generalized mean curvature vector $H_E$ of a Caccioppoli set $E$ is the Radon-Nikodym derivative of $D\mathbf{1}_E$ with respect to $\mathcal{H}^{n-1}$ restricted to $\partial^*E$
  • $H_E$ measures the curvature of the boundary in a weak sense
  • Example: For a smooth set with boundary, $H_E$ coincides with the classical mean curvature vector

Structure theorem for Caccioppoli sets

Rectifiability of the Reduced Boundary

• The structure theorem states that the reduced boundary $\partial^*E$ of a Caccioppoli set $E$ is $(n-1)$-rectifiable
  • Rectifiability means $\partial^*E$ can be covered, up to a negligible set, by countably many $C^1$ hypersurfaces
  • This property reveals the geometric structure of the boundary
• The rectifiability is proven using the Besicovitch-Federer projection theorem
  • The theorem shows that the orthogonal projection of $\partial^*E$ onto almost every $(n-1)$-dimensional subspace has zero $(n-1)$-dimensional Lebesgue measure
  • This implies that $\partial^*E$ is "thin" in almost every direction

Perimeter and Hausdorff Measure

• The structure theorem also establishes that $Per(E, \Omega) = \mathcal{H}^{n-1}(\partial^*E \cap \Omega)$ for any open set $\Omega \subset \mathbb{R}^n$
  • The perimeter of $E$ in $\Omega$ equals the $(n-1)$-dimensional Hausdorff measure of the reduced boundary intersected with $\Omega$
  • This equality connects the measure-theoretic and geometric notions of the boundary
• The proof relies on the Radon-Nikodym theorem and the Gauss-Green theorem
  • The Radon-Nikodym theorem guarantees the existence of $H_E$ as the density of $D\mathbf{1}_E$ with respect to $\mathcal{H}^{n-1}$ on $\partial^*E$
  • Approximating $E$ with smooth sets and applying the Gauss-Green theorem leads to the desired equality

Classification of Caccioppoli sets

Regularity of the Reduced Boundary

• Caccioppoli sets can be classified based on the regularity of their reduced boundary $\partial^*E$
  • A Caccioppoli set $E$ is regular if $\partial^*E$ is a $C^1$ hypersurface
  • Regular sets have smooth boundaries and are easier to study
• The regularity of $\partial^*E$ is related to the integrability of the generalized mean curvature vector $H_E$
  • If $H_E \in L^1(\partial^*E, \mathcal{H}^{n-1})$, then $E$ has integrable mean curvature
  • Sets with integrable mean curvature have more regular boundaries

Stationary Sets and Isoperimetric Inequality

• Caccioppoli sets with vanishing mean curvature ($H_E = 0$ $\mathcal{H}^{n-1}$-almost everywhere on $\partial^*E$) are called stationary sets
  • Stationary sets are critical points of the perimeter functional
  • They play a crucial role in the study of minimal surfaces
• The isoperimetric inequality for Caccioppoli sets states that among all sets with a given volume, balls have the least perimeter
  • Sets that achieve equality in the isoperimetric inequality are called isoperimetric sets
  • Isoperimetric sets have the optimal shape for minimizing perimeter while enclosing a fixed volume

Caccioppoli sets vs Minimal surfaces

Minimal Surfaces and the Plateau Problem

• Minimal surfaces are surfaces that locally minimize their area while being constrained by a given boundary
  • In geometric measure theory, minimal surfaces are closely related to Caccioppoli sets with vanishing mean curvature
  • Stationary Caccioppoli sets can be seen as a generalization of minimal surfaces
• The Plateau problem seeks a surface of least area spanning a given boundary curve
  • The problem can be formulated and solved using Caccioppoli sets
  • The solution to the Plateau problem is a Caccioppoli set with vanishing mean curvature
  • Example: The soap film spanning a wire frame is a physical realization of a minimal surface

Regularity Theory and the Bernstein Problem

• The regularity theory for minimal surfaces can be developed using the properties of Caccioppoli sets
  • The De Giorgi-Allard regularity theorem states that if a Caccioppoli set $E$ has integrable mean curvature and satisfies a density condition, then $\partial^*E$ is a $C^1$ hypersurface
  • This theorem provides conditions under which a minimal surface is smooth
• The Bernstein problem asks whether entire minimal graphs in $\mathbb{R}^n$ are necessarily hyperplanes
  • The problem can be studied using the theory of Caccioppoli sets
  • The De Giorgi-Bernstein theorem provides a positive answer to the Bernstein problem for $n \leq 7$
  • In higher dimensions, there exist non-trivial entire minimal graphs (Bombieri-De Giorgi-Giusti example)