Coarea formula

2010 Mathematics Subject Classification: Primary: 49Q15 [MSN][ZBL]

The coarea formula is a far-reaching generalization of Fubini's theorem in the Euclidean space using "curvilinear coordinates" or "distorted foliations". In what follows we denote by $\mathcal{H}^n$ the $n$-dimensional Hausdorff measure in $\mathbb R^m$ and by $\lambda$ the Lebesgue measure in any $\mathbb R^k$.

Lipschitz maps

Consider a Lipschitz map $f: \mathbb R^m \to \mathbb R^n$, where $m\geq n$. Recall that, by Rademacher's theorem, $f$ is differentiable $\lambda$-a.e.. At any point $y\in \mathbb R^n$ of differentiability we denote by $J f (y)$ the Jacobian of $f$ in $y$, that is the square root of the determinant of $Df|_y \cdot Df|_y^t$ (which, by the Cauchy Binet formula, equals the sum of the squares of the determinants of all $n\times n$ minors of the Jacobian matrix $Df|_y$, see Jacobian).

Theorem 1 Let $f: \mathbb R^m \to \mathbb R^n$ be a Lipschitz function. Then for each $\lambda$-measurable $A\subset \mathbb R^m$, we have

(a) The map $y\mapsto J f (y)$ is Lebesgue measurable;

(b) The set $A\cap f^{-1} (z)$ is an $m-n$-dimensional rectifiable set for $\lambda$-a.e. $z\in \mathbb R^n$;

(c) The map $z\mapsto \mathcal{H}^{m-n} (A\cap f^{-1} (\{z\}))$ is Lebesgue measurable;

(d) The following formula holds \begin{equation}\label{e:coarea} \int_A Jf (y)\, dy = \int_{\mathbb R^n} \mathcal{H}^{m-n} (A\cap f^{-1} (\{z\})\, dz\, . \end{equation}

Cp. with Theorem 1 of Section 3.4.2 of [EG] concerning the points (a), (c) and (d). For point (b) see Theorem 2.93 of [AFP]. The statement of Theorem 1 can be considerably generalized: in particular one can consider Lipschitz maps $f: A \to E$ where $A$ is an $m$-dimensional rectifiable subset of $\mathbb R^M$ and $E$ is an $n$-dimensional rectifiable subset of $\mathbb R^N$. In this case the Jacobian must be suitably defined using an appropriate concept of tangential differentiation. We refer the reader to Theorem 3.2.22 of [Fe].

Fubini type statement

A relatively simple corollary of Theorem 1 is given by the following more general statement, which is also often referred to as Coarea formula.

Theorem 2 Let $f$ be as in Theorem 1 and $g: \mathbb R^m\to \mathbb R$ a $\lambda$-summable function. Then the map $g|_{f^{-1} \{z\}}$ is $\mathcal{H}^{m-n}$ summable for $\lambda$-a.e. $z\in \mathbb R^n$ and the following formula holds \begin{equation}\label{e:coarea2} \int_{\mathbb R^m} g (y)\, Jf (y)\, dy = \int_{\mathbb R^n} \int_{f^{-1} (\{z\})} g(w)\, d\mathcal{H}^{m-n} (w)\, dz\, . \end{equation}

Cp. with Theorem 2 of Section 3.4.3 of [EG]. Observe that when $f$ is the projection onto the first $n$ coordinates, \eqref{e:coarea2} reduces to the classical Fubini theorem.

Relation to Sard's theorem

Assume $f$ in Theorem 1 is of class $C^r$ for some $r> m-n$. Then by Sard's theorem we conclude that $\lambda$-a.e. $z\in \mathbb R^n$ is a regular value of $f$ and hence that $f^{-1} (z)$ is a $C^r$ $m-n$-dimensional submanifold of $\mathbb R^m$. Since $C^1$ submanifolds are rectifiable sets, this implies conclusion (b) in Theorem 1. Thus Theorem 1 can also be considered as an appropriate generalization of Sard's theorem.

BV functions

In case $n=1$ the Lipschitz regularity of $f$ can be considerably relaxed leading to the Fleming-Rishel coarea formula for functions of bounded variation: compare to the Section Coarea formula therein.