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 R^''n'' 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 , and for ' functions by .

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

:$\int_\Omega g(x) , \nabla u(x), \, dx = \int_ \left(\int_g(x)\,dH_(x)\right)\,dt$

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

:$\int_\Omega , \nabla u, = \int_^\infty H_(u^(t))\,dt,$

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

:$\int_\Omega g(x) , J_k u(x), \, dx = \int_ \left(\int_g(x)\,dH_(x)\right)\,dt$

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

:$, J_k u(x), = \left(\right)^.$

Applications

* Taking ''u''(''x'') = , ''x'' − ''x''₀, gives the formula for integration in spherical coordinates of an integrable function ''f'':
::$\int_f\,dx = \int_0^\infty\left\\,dr.$
* Combining the coarea formula with the isoperimetric inequality gives a proof of the Sobolev inequality for ''W''^1,1 with best constant:
::$\left(\int_ , u, ^\right)^\le n^\omega_n^\int_, \nabla u,$
:where $\omega_n$ is the volume of the unit ball in $\R^n.$

See also

* Sard's theorem
* Smooth coarea formula

References

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*{{citation, last1=Malý, first1=J, last2=Swanson, first2=D, last3=Ziemer, first3=W, title=The co-area formula for Sobolev mappings, journal=Transactions of the American Mathematical Society, year=2002, volume=355, pages=477–492, url=https://www.ams.org/tran/2003-355-02/S0002-9947-02-03091-X/S0002-9947-02-03091-X.pdf, format=PDF, doi=10.1090/S0002-9947-02-03091-X, issue=2, doi-access=free.

Measure theory