Saint-Venant's theorem

In solid mechanics, it is common to analyze the properties of beams with constant cross section. Saint-Venant's theorem states that the simply connected cross section with maximal torsional rigidity is a circle. E. Makai, A proof of Saint-Venant's theorem on torsional rigidity, Acta Mathematica Hungarica, Volume 17, Numbers 3-4 / September, 419-422,1966DOI|10.1007/BF01894885 ] It is named after the French mathematician Adhémar Jean Claude Barré de Saint-Venant.

Given a simply connected domain "D" in the plane with area "A" , $ho$ the radius and $sigma$ the area of its greatest inscribed circle, the torsional rigidity "P" of "D" is defined by:$P=\; mathrm\{sup\}\_f\; frac\{left(\; intintlimits\_D\; f,\; dx,\; dy\; ight)^2\}\{intintlimits\_D\; f\_x^2+f\_y^2,\; dx,\; dy\}$

Saint-Venant Saint-Venant [A J-C Barre de Saint-Venant, Memoire sur la torsion des prismes, Memoires presentes par divers savants a l'Acaddmie des Sciences, 14 (1856), pp. 233--560.] conjectured in 1856 thatof all domains "D" of equal area "A" the circular one has the greatest torsional rigidity, that is:$P\; le\; P\_\{mathrm\{circle\; le\; frac\{A^2\}\{2\; pi\}$ a rigorous proof of this inequality was not given until 1948 by Polya [G. Polya, Torsional rigidity, principal frequency, electrostatic capacity and symmetrization, Quarterly of Applied Math., 6 (1948), pp. 267 277.] . Another proof was given by Davenport and reported in [G. Polya and G. Szego, Isoperimetric inequalities in Mathematical Physics (Princeton Univ.Press, 1951).] . A more general proof and an estimate :

is given by Makai

References