On the Gradient of the Torsion Function for Elongated Cylinders

Let $(0,L)\times E_a\subset R^{d+1}$ denote the cylinder of length L, and base $E_a\subset R^d,$ where $E_a$ is an open ellipsoid with semi-axes $a=(a_1,...,a_d)$ . Let $w_{(0,L)\times E_a}$ denote its torsion function. Heat equation tools are used to show that (i) if $E_a$ has small eccentricity and L is sufficiently large, then the maxima of $|\nabla w_{(0,L)\times E_a}|$ 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.