Minimal rearrangements of Sobolev functions. (English) Zbl 0633.46030

Brothers, John E.; Ziemer, William P.

J. Reine Angew. Math. 384, 153-179 (1988).

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

MSC:

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.)

Keywords:

spherically symmetric rearrangement