Isoperimetric Inequalities in the Anisotropic Rod Torsion Problem*

摘要

This paper concerns the optimization problem of finding the best shape of a twisted elastic linearly anisotropic bar, to minimize the maximum elastic energy density over the cross section, with the cross-sectional area specified. Two isoperimetric inequalities for maximal elastic energy density are stated. It is proved that among all solid bars with the same cross-sectional area, the bar with constant density of elastic energy along the boundary possesses the minimum of the maximal value of elastic energy density, with fixed torque. This means that the shapes of rods with the maximal torsional rigidity and minimal concentration of elastic energy density are the same. It is also proved that among all multiply connected regions with fixed outer contour, the best shape is the region that is bounded by two similar ellipses.