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).

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

[math]\displaystyle{ \int_\Omega g(x) |\nabla u(x)|\, dx = \int_{\R} \left(\int_{u^{-1}(t)}g(x)\,dH_{n-1}(x)\right)\,dt }[/math]

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

[math]\displaystyle{ \int_\Omega |\nabla u| = \int_{-\infty}^\infty H_{n-1}(u^{-1}(t))\,dt, }[/math]

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 [math]\displaystyle{ \Omega \subset \R^n, }[/math] taking on values in [math]\displaystyle{ \R^k }[/math] where k ≤ n. In this case, the following identity holds

[math]\displaystyle{ \int_\Omega g(x) |J_k u(x)|\, dx = \int_{\R^k} \left(\int_{u^{-1}(t)}g(x)\,dH_{n-k}(x)\right)\,dt }[/math]

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

[math]\displaystyle{ |J_k u(x)| = \left({\det\left(J u(x) J u(x)^\intercal\right)}\right)^{1/2}. }[/math]

Applications

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

[math]\displaystyle{ \int_{\R^n}f\,dx = \int_0^\infty\left\{\int_{\partial B(x_0;r)} f\,dS\right\}\,dr. }[/math]

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

[math]\displaystyle{ \left(\int_{\R^n} |u|^{\frac{n}{n-1}}\right)^{\frac{n-1}{n}}\le n^{-1}\omega_n^{-\frac{1}{n}}\int_{\R^n}|\nabla u| }[/math]

where [math]\displaystyle{ \omega_n }[/math] is the volume of the unit ball in [math]\displaystyle{ \R^n. }[/math]

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 .
• Federer, Herbert (1959), "Curvature measures", Transactions of the American Mathematical Society (Transactions of the American Mathematical Society, Vol. 93, No. 3) 93 (3): 418–491, doi:10.2307/1993504 .
• 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, https://www.ams.org/tran/2003-355-02/S0002-9947-02-03091-X/S0002-9947-02-03091-X.pdf .