A method of symmetrizing functions and its application to certain problems in elasticity theory for non-uniform bodies.

*(English. Russian original)*Zbl 0724.73046
J. Appl. Math. Mech. 52, No. 3, 377-382 (1988); translation from Prikl. Mat. Mekh. 52, No. 3, 486-492 (1988).

A symmetrization operation is introduced for functions defined in bounded domains and vanishing on the boundary. The properties of the operation introduced are studied and its connection with Schwarz symmetrization is analyzed. Examples are considered of the application of the apparatus developed for constructing isoperimetric estimates in problems of torsion and longitudinal vibrations of an inhomogeneous rod. The stiffness estimate obtained in the problem of the torsion of a nonuniform rod is a generalization of the Polya isoperimetric inequality known in the theory of elasticity for the stiffness of a uniform rod under torsion.

##### MSC:

74B99 | Elastic materials |

74H45 | Vibrations in dynamical problems in solid mechanics |

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74P10 | Optimization of other properties in solid mechanics |

49Q20 | Variational problems in a geometric measure-theoretic setting |

##### Keywords:

symmetrization operation; Schwarz symmetrization; isoperimetric estimates; torsion; longitudinal vibrations; inhomogeneous rod; stiffness estimate
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\textit{S. R. Brudnyĭ} and \textit{E. I. Shifrin}, J. Appl. Math. Mech. 52, No. 3, 377--382 (1988; Zbl 0724.73046); translation from Prikl. Mat. Mekh. 52, No. 3, 486--492 (1988)

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##### References:

[1] | Polya, G.; Szegö, G., Isoperimetric inequalities in mathematical physics, (1962), Fizmatgiz Moscow, /Russian translation/ · Zbl 0101.41203 |

[2] | Beesack, P.R.; Schwarz, B., On the zeros of solution of second-order linear differential equations, Canad. J. math., 8, 4, (1956) · Zbl 0074.30602 |

[3] | Schwarz, B., Bounds for the principal frequency of the non-homogeneous membrane and the generalized Dirichlet integral, Pacif. J. math., 7, 4, (1957) |

[4] | Shifrin, E.I., Estimates of the solution of the problem of a plane mode-one crack in a material with power-law hardening, (), 4 |

[5] | Lomakin, V.A., Theory of elasticity of non-uniform bodies, (1976), Izd. Mosk. Gos. Univ Moscow |

[6] | Panovko, Ya.G., Principles of the applied theory of vibrations and impact, (1976), Mashinostroyenie Leningrad · Zbl 0025.11204 |

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