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Minimal rearrangements of Sobolev functions. (English) Zbl 0633.46030
Let \(A: {\mathbb{R}}^+\to {\mathbb{R}}^+\) be \(C^ 2\) with \(A(0)=0\) and \(A^{1/p}\) convex. Let \(u\geq 0\) be measurable with compact support, and assume \(\nabla u\) is a measurable function with \(\int A(| \nabla u|)<\infty\). Denote by \(u^*\) the spherically symmetric rearrangement of u. Denote \(M=ess \sup u\leq \infty\), \(C=\{x:\nabla u(x)=0\}\). It is proved that \(u,u^*\in W^{1,p}({\mathbb{R}}^ n)\) and \((*)\quad \int A(| Zu^*|)\leq \int A(| Zu|).\) Moreover, if equality holds with \(p>1\), \(| C\cap u^{-1}(0,M)| =0\), and A strictly increasing, then \(u^*\) is equal almost everywhere to a translate of u. The analogous result holds on the n-sphere. Examples are constructed where \(u,u^*\in C^{\infty}\), \(| C\cap u^{- 1}(0,M)| \neq 0\), A is strictly increasing with \(p>1\), and equality holds in (*), but u is not a translate of \(u^*\).
Reviewer: J.E.Brothers

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26D10 Inequalities involving derivatives and differential and integral operators
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Full Text: Crelle EuDML