On the gradient of the torsion function for elongated cylinders

2026/07/20 by Michiel van den Berg
#math.AP

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Abstract

Let (0,L)× E_a⊂ \R^d+1 denote the cylinder of length L, and base E_a⊂ \R^d, where E_a is an open ellipsoid with semi-axes a=(a₁,...,a_d). Let w_{(0,L)× Ea} denote its torsion function. Heat equation tools are used to show that (i) if E_a is has small eccentricity and L is sufficiently large, then the maxima of |∇ w_{(0,L)× Ea}| are located at the centres (0,0) and (L,0) respectively, (ii) if E_a has large eccentricity and L is sufficiently large, then the maxima are located on the lateral side of the cylinder.

Citations

• Hardy inequality and L p estimates for the torsion function
• Estimates for Dirichlet Eigenfunctions
• An upper bound for the first zero of Bessel functions
• Remark on the preceding paper of Serrin

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