Volume-dependent geometry and symmetry transitions of torsion optimizers in infinite cylinders

Abstract

We study relative torsional rigidity in infinite cylinders, addressing two geometric questions raised by Caldiroli, Iacopetti and Pacella on the shape of volume-constrained optimizers. The torsion problem has Dirichlet conditions on the free boundary and Neumann conditions on the fixed lateral wall. At small volumes we establish a two-term expansion of the optimal energy, curvature selection, and quantitative convergence of optimizers to boundary half-balls. We also obtain a uniform expansion for the exact torsion states on the explicit curved half-ball comparison domains. At sufficiently large volumes, every optimizer is exactly a finite cylinder, without convexity of the cross-section or an a priori free-boundary regularity assumption. The normalized optimal value is strictly decreasing up to a finite global transition threshold and constant thereafter. We distinguish this threshold from the local spectral stability threshold and prove coexistence of cylindrical and noncylindrical optimizers when the thresholds are separated. Exact one-sided derivative formulas identify nondifferentiability with the range of transverse energies among optimal states. Strict separation holds for all rectangular cross-sections and for sufficiently close smooth convex inner approximations of any rectangle, with no aspect-ratio restriction. These smooth examples exhibit both curvature-dependent small-volume concentration and a global symmetry transition.