Continuous symmetrization via polarization

TL;DR

This paper introduces a continuous transformation framework for (k,n)-Steiner symmetrizations, enabling monotonic variation of sets and functions, and deriving continuous inequalities and comparison theorems for PDEs.

Contribution

It develops a novel continuous symmetrization process for (k,n)-Steiner symmetrizations, extending previous discrete methods and applying to inequalities and PDE comparison theorems.

Findings

Provides continuous paths for symmetrizations with monotonic properties

Derives continuous convolution and Dirichlet inequalities

Establishes comparison theorems for elliptic and parabolic PDEs

Abstract

We discuss a one-parameter family of transformations which 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 in this stage rely on a simple rearrangement called polarization. In 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\leq k \leq n. This transformation provides us with the desired continuous path, along which all basic characteristics of sets and functions vary monotonically. The latter 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.