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

Abstract

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 analysed. Examples are considered of the application of the apparatus developed for constructing isoperimetric estimates in problems of the torsion and longitudinal vibrations of an inhomogeneous rod. The stiffness estimate obtained in the problem of the torsion of a non-uniform rod is a generalization of the Polya isoperimetric inequality known in the theory of elasticity for the stiffness of a uniform rod under torsion. The solution of many problems in the theory of elasticity encounters serious mathematical difficulties. Nevertheless, it is not so much the stress and displacement fields that are often of practical interest as are certain of their integral characteristics (for instance, the stiffness of an elastic rod under torsion, the frequency of the fundamental of the natural vibrations of a membrane, the first critical force of a compressed rod, etc.). In a number of cases they have been successfully estimated without finding the complete solution of the problem. Among all the possible estimates, the most effective ones are the isoperimetric estimates in which the desired quantity is estimated in terms of the appropriate characteristic of the solution of the simpler problem that allows an analytic or effective numerical solution. The construction of isoperimetric inequalities for the solutions of boundary value problems is based, as a rule, on the application of Steiner or Schwarz symmetrization operations for function level lines /1/ which retain the L p -norm of the functions and do not magnify the corresponding norm of its gradient. This apparatus turns out to be effective for constructing estimates of solutions of a certain class of differential equations with constant coefficients /1/ and variable coefficients in the smallest terms /2, 3/. Utilization of Steiner and Schwarz symmetrization also enables isoperimetric inequalities to be obtained for a certain type of pseudodifferential equation /4/. However, the methods developed in /1–4/ do not enable estimates to be obtained for solutions of boundary value problems for differential equations with variable coefficients in the highest derivatives, and it is such equations that are encountered in elasticity theory problems for non-uniform bodies. The reason for the difficulties occurring here is that the L p -norm must be estimated for the gradient containing weight functions. Standard symmetrization operations of function level lines transform its gradient in an arbitrary manner, whereupon effective reconstruction of the weight function in the symmetrized domain is not successful and, therefore, neither is the required estimate. Below we propose a new symmetrization operation that enables the above-mentioned difficulty to be overcome and examples of its application are examined.