Coarea formula

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

2020 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.


[EG] L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[Fe] H. Federer, "Geometric measure theory", Springer-Verlag (1979). MR0257325 Zbl 0874.49001
[Ma] P. Mattila, "Geometry of sets and measures in euclidean spaces". Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005
[Si] L. Simon, "Lectures on geometric measure theory", Proceedings of the Centre for Mathematical Analysis, 3. Australian National University. Canberra (1983) MR0756417 Zbl 0546.49019
How to Cite This Entry:
Coarea formula. Encyclopedia of Mathematics. URL: