The torsion function of a convex planar domain Ω has convex level sets, but explicit formulae are known only for rectangles and ellipses. Here we study the torsion function when the eccentricity of the domain is large. We obtain an approximation for the torsion function on any convex planar domain by viewing the domain as a perturbation of a rectangle in order to define an approximate Green’s function for the Laplacian. For a class of convex domains we use this approximation to establish sharp bounds on the Hessian and the infinitesimal shape of the level sets around its maximum. We also use these results to construct examples demonstrating contrasting behaviour of the torsion function and the first eigenfunction of the Dirichlet Laplacian around their respective maxima.

中文翻译：

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高偏心凸域的扭转函数

凸平面域Ω的扭转函数具有凸水平集，但是只有矩形和椭圆形才有明确的公式。在这里，我们研究当域的偏心率大时的扭转函数。通过将区域视为矩形的摄动，我们可以得出任何凸平面域上的扭转函数的近似值，以便为拉普拉斯算子定义一个近似格林函数。对于一类凸域，我们使用这种近似在Hessian上建立尖锐的边界，并且级别的无穷小形状围绕其最大值设置。我们还使用这些结果来构造示例，以证明扭转函数和Dirichlet拉普拉斯算子的第一本征函数在其各自最大值附近的对比行为。