7.1 Sets of finite perimeter and the Gauss-Green theorem

Sets of finite perimeter are crucial in geometric measure theory. They extend smooth boundaries to include irregular shapes, allowing for a broader application of key theorems. This concept bridges the gap between classical and modern geometric analysis.

The Gauss-Green theorem for these sets is a powerful tool. It connects the flux of a vector field through a set's boundary to its divergence inside, even for non-smooth boundaries. This opens up new possibilities in studying minimal surfaces and variational problems.

Sets of Finite Perimeter

Definition and Properties

• A set $E \subset \mathbb{R}^n$ has finite perimeter if the distributional gradient of its characteristic function $1_E$ is a vector-valued Radon measure with finite total variation
• The perimeter of $E$, denoted by $P(E)$, is defined as the total variation of the distributional gradient of $1_E$
• If $E$ has a smooth boundary, then $P(E)$ equals the $(n-1)$-dimensional Hausdorff measure of the topological boundary $\partial E$ (smooth manifolds)
• The reduced boundary $\partial^* E$ of a set $E$ of finite perimeter is a subset of the topological boundary $\partial E$, and it is countably $(n-1)$-rectifiable (countable union of Lipschitz graphs)
• The perimeter of $E$ can be expressed as the $(n-1)$-dimensional Hausdorff measure of the reduced boundary $\partial^* E$

Isoperimetric Inequality and Convergence

• Sets of finite perimeter satisfy the isoperimetric inequality, which relates the perimeter and the volume of the set (unit ball in $\mathbb{R}^n$)
• The class of sets of finite perimeter is closed under local convergence in measure, and the perimeter is lower semicontinuous with respect to this convergence
• Example: a sequence of sets with increasingly irregular boundaries converging to a set with a smooth boundary
• Example: approximating a set of finite perimeter by a sequence of smooth sets using the Morse-Sard theorem and the coarea formula

Gauss-Green Theorem for Sets of Finite Perimeter

Statement and Significance

• The Gauss-Green theorem, also known as the divergence theorem, relates the flux of a vector field through the boundary of a set to the divergence of the vector field inside the set
• For a set $E$ of finite perimeter and a smooth vector field $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$, the Gauss-Green theorem states that $\int_E \mathrm{div} F dx = \int_{\partial^* E} F \cdot \nu \mathcal{H}^{n-1}$, where $\nu$ is the measure-theoretic outer unit normal to $E$
• The Gauss-Green theorem for sets of finite perimeter generalizes the classical divergence theorem to a wider class of sets, including those with non-smooth boundaries (fractals, domains with cusps)
• Example: computing the flux of a constant vector field through the boundary of a unit cube in $\mathbb{R}^3$

Proof Outline

• First, approximate $E$ by smooth sets using the Morse-Sard theorem and the coarea formula
• Apply the classical Gauss-Green theorem to the approximating sets and pass to the limit using the Reshetnyak continuity theorem
• The proof relies on the Radon-Nikodym theorem and the properties of the reduced boundary
• Example: approximating a set with a fractal boundary by a sequence of smooth sets and applying the Gauss-Green theorem to each approximation

Applications of the Gauss-Green Theorem

Perimeter Computation and Variational Problems

• The Gauss-Green theorem can be used to compute the perimeter of a set $E$ by choosing appropriate vector fields $F$
  • For example, choosing $F(x) = x/|x|$ leads to the expression $P(E) = \int_{\partial^* E} \nu \mathcal{H}^{n-1}$, where $\nu$ is the measure-theoretic outer unit normal to $E$
• The theorem can be applied to derive the first variation formula for the perimeter functional, which is useful in studying minimal surfaces and related geometric variational problems (soap films, isoperimetric sets)
• The Gauss-Green theorem can be used to prove the existence of a generalized mean curvature for sets of finite perimeter, defined as the Radon-Nikodym derivative of the perimeter measure with respect to the $(n-1)$-dimensional Hausdorff measure on the reduced boundary

Regularity of Minimizers

• The theorem can be employed to establish the regularity of minimizers of the perimeter functional under certain constraints, such as the isoperimetric problem or the Plateau problem
• Example: proving that the minimizer of the perimeter functional among all sets with a given volume is a ball (in $\mathbb{R}^n$)
• Example: showing that the minimizer of the perimeter functional among all sets with a given boundary curve is a minimal surface (soap film spanning a wire frame)

Sets of Finite Perimeter vs Functions of Bounded Variation

Functions of Bounded Variation

• A function $u: \mathbb{R}^n \rightarrow \mathbb{R}$ is said to be of bounded variation (BV) if its distributional gradient is a vector-valued Radon measure with finite total variation
• The space of functions of bounded variation, denoted by $BV(\mathbb{R}^n)$, is a Banach space when endowed with the norm $\|u\|_{BV} = \|u\|_{L^1} + |Du|(\mathbb{R}^n)$, where $|Du|$ is the total variation of the distributional gradient of $u$
• Example: the characteristic function of a set of finite perimeter is a BV function
• Example: a piecewise constant function with a finite number of jump discontinuities is a BV function

Relationship between Sets of Finite Perimeter and BV Functions

• The sublevel sets $\{u > t\}$ of a BV function $u$ have finite perimeter for almost every $t \in \mathbb{R}$, and the perimeter of $\{u > t\}$ is related to the total variation of $Du$ by the coarea formula
• Conversely, given a family of sets $\{E_t\}_{t \in \mathbb{R}}$ with finite perimeter satisfying certain conditions, one can construct a BV function $u$ such that $\{u > t\} = E_t$ for almost every $t \in \mathbb{R}$
• The Gauss-Green theorem for sets of finite perimeter can be used to prove the Fleming-Rishel formula, which expresses the total variation of a BV function in terms of the perimeters of its sublevel sets
• The study of sets of finite perimeter and BV functions is closely related to the theory of minimal surfaces, image processing, and the calculus of variations (denoising, segmentation, inpainting)