Variational conformally invariant inequalities and their applications (Q1593999)

The author provides the solution to two old problems formulated by \textit{B. de Saint-Venant} [[Mémoire présentées par divers savants à l'Academie des Sciences 14, 233-560 (1856)] and \textit{Lord Rayleigh} [The theory of sound. Vol. I, II (1894; JFM 25.1604.01), (1986; JFM 27.0701.05)]. The problems have a reference to two-sided estimates expressed in terms of geometric parameter -- the torsional stiffness -- and a similar problem for the fundamental frequency of a vibrating membrane. The reasonings are based on new inequalities that are conformally invariant analogues of the known Poincaré-Steklov-Friedrichs inequalities.