Solution of the generalized Saint Venant problem

A well-known problem in the mathematical theory of elasticity about the torsional rigidity $P(\Omega)$ of a bar whose cross-section is an arbitrary simply connected domain $\Omega$ is considered. It is shown that $P(\Omega)$ is equivalent to the moment of inertia of the domain relative to its boundary. Thus, a new interpretation of the well-known Coulomb's formula is suggested, and on this basis the following problem, which has its origins in works of Cauchy and Saint Venant, is solved: find a geometric parameter equivalent to the torsional rigidity coefficient of elastic bars with simply connected cross-sections. The proof is based on the definition of the torsional rigidity as the norm of a certain embedding operator in a Sobolev space and on the theory of conformal maps. In particular, some conformally invariant inequalities are established.