Computer-assisted local maximality of regular polygons for torsional rigidity

Abstract

We study torsional rigidity as a function of the labeled vertices of a convex polygon. Starting from the distributed second shape derivative, we derive the Hessian with respect to vertex coordinates. At a regular polygon, dihedral symmetry makes this matrix block circulant in radial-tangential coordinates, reducing its spectrum to the eigenvalues of Hermitian matrices of order two. We also derive an exact second-variation Galerkin identity and guaranteed functional residual majorants. Finite elements approximate the PDE solutions entering the Hessian, and FLINT/Arb provides the interval arithmetic needed for certification. In the scale-invariant setting, we certify exactly four zero eigenvalues generated by similarities and $2n-4$ strictly negative eigenvalues for $5\leq n\leq25$. The regular polygons in this range are therefore strict local maximizers, modulo similarities, of torsional rigidity divided by area squared. The observed Hessian error decreases nearly quadratically with the mesh size; the transmission regularity needed to prove this rate is stated separately as a conjecture.