A sharp upper bound for the torsional rigidity of rods by means of web functions

Using web functions, we approximate the Dirichlet integral which represents the torsional rigidity of a cylindrical rod with planar convex cross-section Ω. To this end, we use a suitably defined piercing function, which enables us to obtain bounds for both the approximate and the exact value of the torsional rigidity. When Ω varies, we show that the ratio between these two values is always larger than ¾; we prove that this is a sharp lower bound and that it is not attained. Several extremal cases are also analyzed and studied by numerical methods.