关于三角形和矩形的一个 Pólya 泛函猜想
On a conjecture of a Pólya functional for triangles and rectangles
AI总结：
本文研究了平面区域第一 Dirichlet 特征值与扭转刚度乘积的 Pólya 泛函，证明了其在三角形和矩形类中下界为 $π^{2}/24$、上界为 $π^{2}/12$ 的猜想，并验证了矩形的单调性及切线四边形的上界。
摘要：
我们考虑由平面区域的第一 Dirichlet 特征值与扭转刚度乘积除以面积归一化所得的泛函。该尺度不变泛函曾由 Pólya 和 Szegő 于 1951 年研究，他们证明对所有区域其上界为 1。有人猜想，在有界凸平面区域类中，该泛函下界为 $π^{2}/24$，上界为 $π^{2}/12$，且这些界是紧的。值得注意的是，即使在三角形类中，该猜想仍未解决。本文旨在证明此情形下的猜想。该猜想在矩形情形下也得到证明，并验证了更强的单调性。最后，上界对切线四边形也成立。
英文摘要：
We consider the functional given by the product of the first Dirichlet eigenvalue and the torsional rigidity of planar domains normalized by the area. This scale invariant functional was studied by Pólya and Szegő in 1951 who showed that it is bounded above by 1 for all domains. It has been conjectured that within the class of bounded convex planar domains the functional is bounded below by $π^{2}/24$ and above by $π^{2}/12$ and that these bounds are sharp. Remarkably, the conjecture remains open even within the class of triangles. The purpose of this paper is to prove the conjecture in this case. The conjecture is also proved for rectangles where a stronger monotonicity property is verified. Finally, the upper bound also holds for tangential quadrilateral.
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