Coarea formula

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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 the integral of 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 simple 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 Rn, 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_{-\infty}^\infty \left(\int_{u^{-1}(t)}g(x)\,dH_{n-1}(x)\right)\,dt

where Hn − 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 Ω ⊂ Rn, taking on values in Rk 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 Jku is the k-dimensional Jacobian of u.

Applications[edit]

  • Taking u(x) = |x − x0| gives the formula for integration in spherical coordinates of an integrable function ƒ:
\int_{\mathbb{R}^n}f\,dx = \int_0^\infty\left\{\int_{\partial B(x_0;r)} f\,dS\right\}\,dr.
\left(\int_{\mathbb{R}^n} |u|^{n/(n-1)}\right)^{\frac{n-1}{n}}\le n^{-1}\omega_n^{-1/n}\int_{\mathbb{R}^n}|\nabla u|
where ωn is the volume of the unit ball in Rn.

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References[edit]