Mixed Torsion on Right Triangles and the P\'olya--Szeg\H{o} Monotonicity Problem for Regular Polygons

Abstract

Motivated by the polygonal P\'olya--Szeg\H{o} conjecture for torsional rigidity, we study two monotonicity problems for torsional rigidity. The first concerns a mixed torsion problem on fixed-area right triangles, with a Dirichlet condition on one leg and Neumann conditions on the other leg and on the hypotenuse. We prove that the mixed torsional rigidity strictly increases as the ratio of the Neumann leg to the Dirichlet leg increases. The proof uses a Hadamard shape derivative, a Pohozaev-type identity, and a monotonicity result for the mixed torsion function. We also prove a similar result for the mixed ground state of Laplacian. The second concerns regular polygons. If \(P_N\) denotes the regular \(N\)-gon of area \(\pi\), we prove, by a purely analytic Schwarz--Christoffel/Bergman analytic-content argument, that \[ T^D(P_{N+1})>T^D(P_N),\qquad N\ge3, \] where \(T^D\) is the Dirichlet torsional rigidity. We also obtain the asymptotic expansion \[ T^D(P_N)=\frac{\pi}{8}-\frac{\pi\zeta(3)}{N^3} +\frac{\pi^5}{45N^4}+O(N^{-5}). \]