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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

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.)
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