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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Symmetrization with respect to a measure

Authors: Friedmar Schulz and Virginia Vera de Serio
Journal: Trans. Amer. Math. Soc. 337 (1993), 195-210
MSC: Primary 49Q15; Secondary 26B99, 28A20, 30C20
MathSciNet review: 1088477
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Abstract: In this paper we study the spherical symmetric rearrangement $ {u^\ast}$ of a nonnegative measurable function $ u$ on $ {\mathbb{R}^n}$ with respect to a measure given by a nonhomogeneous density distribution $ p$. Conditions on $ u$ are given which guarantee that $ {u^\ast}$ is continuous, of bounded variation, or absolutely continuous on lines, i.e., Sobolev regular. The energy inequality is proven in $ n = 2$ dimensions by employing a Carleman type isoperimetric inequality if $ \log p$ is subharmonic. The energy equality is settled via a reduction to the case of a homogeneous mass density.

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Keywords: Schwarz symmetrization, spherical symmetric rearrangement, isoperimetric inequality, energy inequality, Faber-Krahn inequality, nonhomogeneous mass density, coarea formula, Sobolev function, function of bounded variation
Article copyright: © Copyright 1993 American Mathematical Society