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.
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Communicated by Vladimir V. Andrievskii.
In the memory of Stephan Ruscheweyh, a wonderful person and an excellent mathematician whose work for the mathematical society inspired so many of us.
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Solynin, A.Y. Exercises on the Theme of Continuous Symmetrization. Comput. Methods Funct. Theory 20, 465–509 (2020). https://doi.org/10.1007/s40315-020-00331-y
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DOI: https://doi.org/10.1007/s40315-020-00331-y
Keywords
- Continuous symmetrization
- Torsional rigidity
- Principal frequency
- Logarithmic capacity
- Convex domain
- Triangular beam
- Triangular drum
- Isoperimetric inequality