Abstract
Let F:(0, ∞) × [0, ∞) → R be a function of Carathéodory type. We establish the inequality ∫ R N F ( | x | , u ( x ) ) d x ⩽ ∫ R N F ( | x | , u ∗ ( x ) ) d x . where u* 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 RN is replaced by a set of finite measure. The identity ∫ R N G ( u ( x ) ) d x = ∫ R N G ( u ∗ ( x ) ) d x is also discussed under the assumption that G: [0,∞) → R is a Borel function. 2000 Mathematics Subject Classification 26D20, 42C20, 46E30.
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