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Transactions of the American Mathematical Society

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



An approach to symmetrization
via polarization

Authors: Friedemann Brock and Alexander Yu. Solynin
Journal: Trans. Amer. Math. Soc. 352 (2000), 1759-1796
MSC (1991): Primary 28D05, 58G35, 35A30, 35B05, 35B50, 35J60, 35K55, 26D10
Published electronically: December 10, 1999
MathSciNet review: 1695019
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Abstract: We prove that the Steiner symmetrization of a function can be approximated in $L^p ({\mathbb R}^n )$ by a sequence of very simple rearrangements which are called polarizations. This result is exploited to develop elementary proofs of many inequalities, including the isoperimetric inequality in Euclidean space. In this way we also obtain new symmetry results for solutions of some variational problems. Furthermore we compare the solutions of two boundary value problems, one of them having a "polarized" geometry and we show some pointwise inequalities between the solutions. This leads to new proofs of well-known functional inequalities which compare the solutions of two elliptic or parabolic problems, one of them having a "Steiner-symmetrized" geometry. The method also allows us to investigate the case of equality in the inequalities. Roughly speaking we prove that the equality sign is valid only if the original problem has the symmetrized geometry.

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Additional Information

Friedemann Brock
Affiliation: Mathematisches Institut, Universität Köln, Weyertal 90, D 50923 Köln, Germany
Address at time of publication: Department of Mathematics, University of Missouri-Columbia, Mathematical Sciences Building, Columbia, Missouri 65211

Alexander Yu. Solynin
Affiliation: Mathematisches Institut, Universität Köln, Weyertal 90, D 50923 Köln, Germany; Russian Academy of Sciences, V.A. Steklov Mathematical Institute, St. Petersburg Branchm, Fontanka 27, 191011 St. Petersburg, Russia

Keywords: Steiner symmetrization, rearrangement, polarization, integral inequality, boundary value problem, comparison theorem, maximum principle, uniqueness theorem
Received by editor(s): May 15, 1996
Published electronically: December 10, 1999
Additional Notes: Research supported by Volkswagen-Stiftung, RiP-program
Article copyright: © Copyright 2000 American Mathematical Society

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