Old and New on the Bass Note, the Torsion Function and the Hyperbolic Metric

A cylindrical beam of uniform cross section is subject to an infinites-imal torsion. One needs to know the resulting stresses. The theory of elasticity reduces this to the solution of the problem{ ∆u = −2 in D, u = 0 on ∂D, (0.1) if the beam in three dimensional Cartesian coordinates is D × R where the cross section D is a simply connected domain in the xy-plane. The partial derivatives uy and −ux give the components in the x and y directions respectively of the stress vector relative to the normal direction to D. The torsional rigidity of the beam is P de