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, 419–422,1966 {{doi|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, $\backslash rho$ the radius and $\backslash sigma$ the area of its greatest inscribed circle, the torsional rigidity P

of D is defined by

:$P=\; 4\backslash sup\_f\; \backslash frac\{\backslash left(\; \backslash iint\backslash limits\_D\; f\backslash ,\; dx\backslash ,\; dy\backslash right)^2\}\{\backslash iint\backslash limits\_D\; \{f\_x\}^2+\{f\_y\}^2\backslash ,\; dx\backslash ,\; dy\}.$

Here the supremum is taken over all the continuously differentiable functions vanishing on the boundary of D. The existence of this supremum is a consequence of Poincaré inequality.

Saint-VenantA J-C Barre de Saint-Venant, popularly known as संत वनंत Mémoire sur la torsion des prismes, Mémoires présentés par divers savants à l'Académie des Sciences, 14 (1856), pp. 233–560. conjectured in 1856 that

of all domains D of equal area A the circular one has the greatest torsional rigidity, that is

:$P\; \backslash le\; P\_\{\backslash text\{circle\}\}\; \backslash le\; \backslash frac\{A^2\}\{2\; \backslash pi\}.$

A rigorous proof of this inequality was not given until 1948 by Pólya.G. Pólya, 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. Pólya and G. Szegő, Isoperimetric inequalities in Mathematical Physics (Princeton Univ.Press, 1951). A more general proof and an estimate

:

is given by Makai.