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A co-area formula with applications to monotone rearrangement and to regularity. (English) Zbl 0735.49039

Rearrangements of functions have been used extensively in the study of problems arising from plasma physics, but perhaps are not widely known. Let \(\Omega\subset \mathbb{R}^ N\) be Lebesgue measurable and of finite measure. The decreasing rearrangement of a measurable function u defined on \(\Omega\) is the function \(u_ *: (0,|\Omega|)\rightarrow \mathbb{R}\) defined by \[ u_ *(s)=\sup\{t\in\mathbb{R}: {\mathcal L}^ N(\{y\in \Omega: u(y)>t\})>s\}. \] There is also defined the more complicated notion of the relative rearrangement of one function with respect to another. Because of the use of quantities such as \({\mathcal L}(\{y\in\Omega: u(y)>t\})\), it should not be surprising that the co-area formula would be useful in considering rearrangements. That connection is explored in this paper and various applications obtained. As an example which will not require additional definitions, it is shown that if u is in the Sobolev space \(W^{1,p}(\Omega)\), then \(u_ *\) is in \(W^{1,p}_{loc}(0,|\Omega|)\).
Reviewer: H.Parks

MSC:

49Q15 Geometric measure and integration theory, integral and normal currents in optimization
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