Calculus of variations and optimal control; optimization Geometric measure theory and optimization on manifolds Theorem

De Giorgi's structure theorem

Builds on Caccioppoli set Perimeter of a set Radon measure Hausdorff measure Rectifiable set

Statement

Let $n \ge 2$ and let $E \subseteq \R^n$ be a set of locally finite perimeter, that is, a local Caccioppoli set. Let $\mu = \abs{D\chi_E}$ be the total variation measure of the distributional gradient of its indicator function $\chi_E$ , a Radon measure. Write $D\chi_E = -\nu_E \, \mu$ with a Borel map $\nu_E \colon \R^n \to \R^n$ such that $\abs{\nu_E} = 1$ $\mu$ -almost everywhere. For $x \in \R^n$ and $r > 0$ let $B_r(x)$ be the open ball of radius $r$ about $x$ . Consider the limit $\lim_{r \to 0} \frac{D\chi_E(B_r(x))}{\mu(B_r(x))}.$ The reduced boundary $\partial^* E$ is the set of points $x$ in the support of $\mu$ at which this limit exists and has norm $1$ . Let $\mathcal{H}^{n-1}$ be the $(n-1)$ -dimensional Hausdorff measure. Then:

The set $\partial^* E$ is countably $(n-1)$ -rectifiable: up to an $\mathcal{H}^{n-1}$ -null set, it is contained in a countable union of $C^1$ hypersurfaces.
$\mu = \mathcal{H}^{n-1} \llcorner \partial^* E$ , that is, $\mu(A) = \mathcal{H}^{n-1}(A \cap \partial^* E)$ for every Borel set $A$ .
For every $x \in \partial^* E$ , the rescaled sets $(E - x)/r$ converge in $L^1_{\mathrm{loc}}(\R^n)$ , as $r \to 0$ , to the half-space $\set{y : y \cdot \nu_E(x) < 0}$ .
For every $\varphi \in C^1_c(\R^n; \R^n)$ , the generalized Gauss–Green formula holds: $\int_E \operatorname{div} \varphi \, dx = \int_{\partial^* E} \varphi \cdot \nu_E \, d\mathcal{H}^{n-1}.$

Remarks

The proof starts from the Besicovitch differentiation theorem, which shows that $\mu$ -almost every point lies in $\partial^* E$ . At a point of $\partial^* E$ , the isoperimetric inequality gives density estimates comparing $\mu(B_r(x))$ with $r^{n-1}$ . With the compactness of sets of finite perimeter, a blow-up argument then shows that the rescaled sets converge to a half-space. The half-space limit identifies $\mu$ with $\mathcal{H}^{n-1}$ on $\partial^* E$ . Rectifiability then follows from the existence of approximate tangent planes at all points of $\partial^* E$ . For a bounded open set with $C^1$ boundary, $\partial^* E = \partial E$ , $\nu_E$ is the outer unit normal, and item 4 is the classical divergence theorem. For the square $[0, 1]^2$ , the reduced boundary is the boundary minus the four corners; the corners form an $\mathcal{H}^1$ -null set. The topological boundary can be much larger than $\partial^* E$ , even of positive Lebesgue measure, so the reduced boundary is the right notion of boundary here.

Going deeper

For a student

The theorem is due to Ennio De Giorgi, and it is the foundation of the theory of sets of finite perimeter. It turns a set known only through the measure $D\chi_E$ into a geometric object with a measure-theoretic normal. Federer later showed that $\partial^* E$ coincides, up to an $\mathcal{H}^{n-1}$ -null set, with the essential boundary, the set of points where $E$ has density neither $0$ nor $1$ . These results are the starting point of the regularity theory of perimeter minimizers and of the study of free discontinuity problems.

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Tag	`D5VM0MC3EB`
URI	`m2://D5VM0MC3EB`
Kind	statement, theorem
Writing	W0 stub
MSC	49Q15, 28A75, 26B30
Status	This copy of the site has no build data. Run `m2 status D5VM0MC3EB`.
About	Caccioppoli set
Uses	Caccioppoli set, Perimeter of a set, Radon measure, Hausdorff measure, Rectifiable set