Exercises on the Theme of Continuous Symmetrization

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Abstract

In this paper, we discuss the Pólya–Szegő continuous symmetrization and its applications to isoperimetric inequalities. In particular, we survey results concerning monotonicity properties of certain characteristics, including torsional rigidity of cylindrical beams and principal frequency of a uniformly stretched elastic membrane of a drum, of triangles and other domains. Several remaining open problems on continuous symmetrization and relevant properties of domains and functions are also discussed.

Original languageEnglish
Pages (from-to)465-509
Number of pages45
JournalComputational Methods and Function Theory
Volume20
Issue number3-4
DOIs
StatePublished - Nov 2020

Keywords

  • Continuous symmetrization
  • Convex domain
  • Isoperimetric inequality
  • Logarithmic capacity
  • Principal frequency
  • Torsional rigidity
  • Triangular beam
  • Triangular drum

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