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Symmetrization Inequalities for Composition Operators of Carathéodory type

  • H. Hajaiej (a1) and C. A. Stuart (a2)

Abstract

Let $F:(0, \infty) \times [0, \infty) \rightarrow \mathbb{R}$ be a function of Carathéodory type. We establish the inequality $$ \int_{\mathbb{R}^{N}} F( | x |, u(x) ) dx \leq \int_{\mathbb{R}^{N} } F( | x |, u^{\ast}(x)) dx. $$ where $u^{\ast}$ denotes the Schwarz symmetrization of $u$, under hypotheses on $F$ that seem quite natural when this inequality is used to obtain existence results in the context of elliptic partial differential equations. We also treat the case where $\mathbb{R}^N$ is replaced by a set of finite measure. The identity $$ \int_{\mathbb{R}^{N}} G(u(x)) dx = \int_{\mathbb{R}^{N}} G(u^{\ast}(x)) dx $$ is also discussed under the assumption that $G : [0,\infty) \rightarrow \mathbb{R}$ is a Borel function.

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Symmetrization Inequalities for Composition Operators of Carathéodory type

  • H. Hajaiej (a1) and C. A. Stuart (a2)

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