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Symmetrization of functions in Sobolev spaces and the isoperimetric inequality

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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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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

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