“Coarea formula”的意思、由来-开放百科全书

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Coarea formula

释义

Applications
See also
References

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 simple change of variables. More general forms of the formula for Lipschitz functions were first established by Herbert Federer {{harv|Federer|1959}}, and for {{math|BV}} functions by {{harvtxt|Fleming|Rishel|1960}}.

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

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

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 taking on values in where k < n. In this case, the following identity holds

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

Applications

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

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

where is the volume of the unit ball in

References

{{citation|last=Federer|first=Herbert| authorlink = Herbert Federer| title = Geometric measure theory| publisher = Springer-Verlag New York Inc.| location = New York| year = 1969| pages = xiv+676| isbn = 978-3-540-60656-7| mr= 0257325 | series = Die Grundlehren der mathematischen Wissenschaften, Band 153 }}.
{{citation|last=Federer|first=Herbert|authorlink=Herbert Federer|title=Curvature measures|journal=Transactions of the American Mathematical Society|volume=93|year=1959|pages=418–491|jstor=1993504|doi=10.2307/1993504|issue= 3|publisher=Transactions of the American Mathematical Society, Vol. 93, No. 3}}.
{{citation|last1=Fleming|first1=WH|last2=Rishel|first2=R|title=An integral formula for the total gradient variation|journal=Archiv der Mathematik|volume = 11|year=1960|doi=10.1007/BF01236935|pages=218–222|url=http://www.springerlink.com/index/WV67N13464926501.pdf|format=PDF|issue = 1}}
{{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=http://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}}.

1 : Measure theory

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Applications

See also

References