Coarea formula

In the mathematical field of geometric measure theory, the coarea formula expresses the integral of a function over an open set in Euclidean space in terms of integrals over the level sets of another function. A special case is Fubini's theorem, which says under suitable hypotheses that the integral of a function over the region enclosed by a rectangular box can be written as the iterated integral over the level sets of the coordinate functions. Another special case is integration in spherical coordinates, in which the integral of a function on Rⁿ is related to the integral of the function over spherical shells: level sets of the radial function. The formula plays a decisive role in the modern study of isoperimetric problems.

For smooth functions the formula is a result in multivariate calculus which follows from a change of variables. More general forms of the formula for Lipschitz functions were first established by Herbert Federer (Federer 1959), and for BV functions by Fleming & Rishel (1960).

A precise statement of the formula is as follows. Suppose that Ω is an open set in $\mathbb {R} ^{n}$ and u is a real-valued Lipschitz function on Ω. Then, for an L¹ function g,

$\int _{\Omega }g(x)|\nabla u(x)|\,dx=\int _{\mathbb {R} }\left(\int _{u^{-1}(t)}g(x)\,dH_{n-1}(x)\right)\,dt$

where H_n−1 is the (n − 1)-dimensional Hausdorff measure. In particular, by taking g to be one, this implies

$\int _{\Omega }|\nabla u|=\int _{-\infty }^{\infty }H_{n-1}(u^{-1}(t))\,dt,$

and conversely the latter equality implies the former by standard techniques in Lebesgue integration.

More generally, the coarea formula can be applied to Lipschitz functions u defined in $\Omega \subset \mathbb {R} ^{n},$ taking on values in $\mathbb {R} ^{k}$ where k ≤ n. In this case, the following identity holds

$\int _{\Omega }g(x)|J_{k}u(x)|\,dx=\int _{\mathbb {R} ^{k}}\left(\int _{u^{-1}(t)}g(x)\,dH_{n-k}(x)\right)\,dt$

where J_ku is the k-dimensional Jacobian of u whose determinant is given by

$|J_{k}u(x)|=\left({\det \left(Ju(x)Ju(x)^{\intercal }\right)}\right)^{1/2}.$

Applications

• Taking u(x) = |x − x₀| gives the formula for integration in spherical coordinates of an integrable function f:

$\int _{\mathbb {R} ^{n}}f\,dx=\int _{0}^{\infty }\left\{\int _{\partial B(x_{0};r)}f\,dS\right\}\,dr.$

• Combining the coarea formula with the isoperimetric inequality gives a proof of the Sobolev inequality for W^1,1 with best constant:

$\left(\int _{\mathbb {R} ^{n}}|u|^{\frac {n}{n-1}}\right)^{\frac {n-1}{n}}\leq n^{-1}\omega _{n}^{-{\frac {1}{n}}}\int _{\mathbb {R} ^{n}}|\nabla u|$

where $\omega _{n}$ is the volume of the unit ball in $\mathbb {R} ^{n}.$

See also

• Sard's theorem
• Smooth coarea formula

References

• Federer, Herbert (1969), Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, New York: Springer-Verlag New York Inc., pp. xiv+676, ISBN 978-3-540-60656-7, MR 0257325.
• Federer, Herbert (1959), "Curvature measures", Transactions of the American Mathematical Society, 93 (3), Transactions of the American Mathematical Society, Vol. 93, No. 3: 418–491, doi:10.2307/1993504, JSTOR 1993504.
• Fleming, WH; Rishel, R (1960), "An integral formula for the total gradient variation", Archiv der Mathematik, 11 (1): 218–222, doi:10.1007/BF01236935
• Malý, J; Swanson, D; Ziemer, W (2002), "The co-area formula for Sobolev mappings" (PDF), Transactions of the American Mathematical Society, 355 (2): 477–492, doi:10.1090/S0002-9947-02-03091-X.