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 ${\mathbb{ℝ}}^n$ and u is a real-valued Lipschitz function on Ω. Then, for an L¹ function g,

$\underset{\mathrm{\Omega }}{\int }g(x)|\mathrm{\nabla }u(x)|dx=\underset{\mathbb{ℝ}}{\int }(\underset{u^{-1}(t)}{\int }g(x)d{H}_{n-1}(x))dt$

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

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

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 $\mathrm{\Omega }\subset {\mathbb{ℝ}}^n,$ taking on values in ${\mathbb{ℝ}}^k$ where k ≤ n. In this case, the following identity holds

$\underset{\mathrm{\Omega }}{\int }g(x)|J_{k}u(x)|dx=\underset{{\mathbb{ℝ}}^k}{\int }(\underset{u^{-1}(t)}{\int }g(x)d{H}_{n-k}(x))dt$

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

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

Applications

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

$\underset{{\mathbb{ℝ}}^n}{\int }fdx=\underset{0}{\overset{\mathrm{\infty }}{\int }}\left\{\underset{\partial B(x_0;r)}{\int }fdS\right\}dr.$

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

${(\underset{{\mathbb{ℝ}}^n}{\int }|u{|}^{\frac{n}{n-1}})}^{\frac{n-1}{n}}\le n^{-1}{\omega }_n^{-\frac{1}{n}}\underset{{\mathbb{ℝ}}^n}{\int }|\mathrm{\nabla }u|$

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 .
• 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 .