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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Continuous symmetrization via polarization


Author: A. Yu. Solynin
Original publication: Algebra i Analiz, tom 24 (2012), nomer 1.
Journal: St. Petersburg Math. J. 24 (2013), 117-166
MSC (2010): Primary 30C75
Published electronically: November 15, 2012
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Abstract: We discuss a one-parameter family of transformations that changes sets and functions continuously into their $ (k,n)$-Steiner symmetrizations. Our construction consists of two stages. First, we employ a continuous symmetrization introduced by the author in 1990 to transform sets and functions into their one-dimensional Steiner symmetrization. Some of our proofs at this stage rely on a simple rearrangement called polarization. At the second stage, we use an approximation theorem due to Blaschke and Sarvas to give an inductive definition of the continuous $ (k,n)$-Steiner symmetrization for any $ 2\le k \le n$. This transformation provides us with the desired continuous path along which all basic characteristics of sets and functions vary monotonically. In its turn, this leads to continuous versions of several convolution type inequalities and Dirichlet's type inequalities as well as to continuous versions of comparison theorems for solutions of some elliptic and parabolic partial differential equations.


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Additional Information

A. Yu. Solynin
Affiliation: Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, Texas 79409
Email: alex.solynin@ttu.edu

DOI: http://dx.doi.org/10.1090/S1061-0022-2012-01234-3
PII: S 1061-0022(2012)01234-3
Keywords: Continuous symmetrization, Steiner symmetrization, rearrangement, polarization, integral inequality, boundary-value problem, comparison theorem
Received by editor(s): February 7, 2011
Published electronically: November 15, 2012
Article copyright: © Copyright 2012 American Mathematical Society