On the continuity of the Continuous Steiner Symmetrization

TL;DR

This paper explores the process of continuously transforming a domain into a ball while preserving measure and decreasing the Laplace operator's first eigenvalue, building on Brock's construction of Continuous Steiner Symmetrization.

Contribution

It demonstrates that for many cases, such a continuous transformation is possible, advancing understanding of domain symmetrization and eigenvalue optimization.

Findings

Possible to continuously transform domains into balls while decreasing eigenvalues in many cases

Extends Brock's construction of Continuous Steiner Symmetrization

Open questions remain for the general case

Abstract

Starting from the Brock's construction of Continuous Steiner Symmetrization of sets, the problem of modifying continuously a given domain up to obtain a ball, preserving its measure and with decreasing first eigenvalue of the Laplace operator, is considered. For a large class of cases it is shown this is possible, while the general question remains still open.