Abstract
A positive measurable function f on Rd can be symmetrized to a function f* depending only on the distance r, and with the same distribution function as f. If the distribution derivatives of f are Radon measures then we have the inequality ∥∇f*∥≤∥∇f∥, where ∥∇f∥ is the total mass of the gradient. This inequality is a generalisation of the classical isoperimetric inequality for sets. Furthermore, and this is important for applications, if f belongs to the Sobolev space H1,P then f* belongs to H1,P and ∥∇f*∥p≤∥∇f∥p.
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References
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FEDERER, H.: Geometric measure theory. Berlin-Heidelberg-New York: Springer 1969.
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Hildén, K. Symmetrization of functions in Sobolev spaces and the isoperimetric inequality. Manuscripta Math 18, 215–235 (1976). https://doi.org/10.1007/BF01245917
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DOI: https://doi.org/10.1007/BF01245917