关于细长圆柱体扭转函数的梯度
On the gradient of the torsion function for elongated cylinders
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中文总结 AI 辅助
研究关于细长圆柱体扭转函数梯度的问题，借助热方程工具，得出当\(E_a\)离心率小且\(L\)足够大时，\(|\nabla w_{(0,L)\times E_a}|\)最大值在中心；\(E_a\)离心率大且\(L\)足够大时，最大值在圆柱体侧面的结论。
中文摘要 AI 辅助
设\((0,L)\times E_a\subset \R^{d + 1}\)表示长度为\(L\)且底面为\(E_a\subset \R^{d}\)的圆柱体，其中\(E_a\)是半轴为\(a=(a_1,\cdots,a_d)\)的开椭球体，\(w_{(0,L)\times E_a}\)表示其扭转函数。利用热方程工具表明：（i）若\(E_a\)离心率小且\(L\)足够大，则\(|\nabla w_{(0,L)\times E_a}|\)的最大值分别位于中心\((0,0)\)和\((L,0)\)；（ii）若\(E_a\)离心率大且\(L\)足够大，则最大值位于圆柱体侧面。
英文摘要
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 $E_a$ is 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.
