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.

中文翻译：

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以连续对称为主题的练习

在本文中，我们讨论了Pólya–Szegő连续对称化及其在等长不等式中的应用。特别是，我们调查有关某些特性的单调性的结果，包括圆柱梁的扭转刚度以及鼓，三角形和其他区域的均匀拉伸的弹性膜鼓的主频率。还讨论了关于连续对称化以及域和函数的相关属性的几个尚待解决的问题。