Minimal rearrangements of Sobolev functions. (English) Zbl 0659.46029

It is well known that if \(u:{\mathbb{R}}^ n\to {\mathbb{R}}\) is a nonnegative function with compact support, then for \(1\leq p<\infty\), \[ (1)\quad [\int_{{\mathbb{R}}^ n}| \nabla u^*| p]^{1/p}\leq [\int_{{\mathbb{R}}^ n}| \nabla u|^ p]^{1/p} \] where \(u^*\) denotes the spherical symmetric rearrangement of u; \(u^*\) is defined by \[ (2)\quad u^*(x)=\sup \{t:\mu (t)>\alpha (n)| x|^ n\} \] where \(\alpha\) (n) is the volume of the unit n-ball in \({\mathbb{R}}^ n\) and \(\mu (t)<\infty\) is the Lebesgue measure of the set \(E_ t=\{x:u(x)>t\}\). Note that \(\mu (t)=| E^*_ t|\) where \(E_ t^*=\{x:u^*(x)>t\}\) and \(| E^*_ t|\) denotes the Lebesgue measure of \(E^*_ t\). The purpose of this paper is to show that if \(\mu\) is absolutely continuous and equality holds in (1), then u is almost everywhere equal to a translate of \(u^*\). We also construct examples which show that this may not be true if \(\mu\) is not absolutely continuous.


46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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