发表机构
Beijing Normal University（北京师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了二维有界凸域上Pólya泛函的尖锐不等式，即第一Dirichlet特征值与扭转刚度之积除以面积介于π²/24和π²/12之间，从而解决了van den Berg等人提出的猜想，并验证了端点常数的尖锐性。
AI 中文摘要
设$\lambda_1(\Omega)$和$T(\Omega)$分别表示有界凸域$\Omega\subset\mathbb{R}^2$的第一Dirichlet特征值和扭转刚度，并设$M=\max_\Omega u$，其中$u$是扭转函数。我们证明了尖锐不等式$\pi^2/24<\lambda_1(\Omega)T(\Omega)/|\Omega|<\pi^2/12$。这证明了van den Berg、Buttazzo和Pratelli提出的猜想4.2的二维情形。其平面形式后来被Bañuelos和Mariano重述为猜想1.1，他们证明了三角形和矩形的情形。下界通过将Payne关于$\lambda_1M$的严格估计与尖锐的扭转效率不等式$T(\Omega)\geq |\Omega|M/3$相结合而得出。对于光滑的严格凸域，我们使用Airy应力势构造一个凸体，其表面积测度是扭转第一变分测度。一个关于方向最大剖面的尖锐一维不等式随后产生更强的几何包含关系。对于上界，我们将Pólya泛函通过第二扭转矩进行分解。一个水平集扭转-周长不等式给出因子$5/6$，而一个尖锐的加权一维估计给出因子$\pi^2/10$。坍缩三角形和拉长矩形表明两个端点常数以及效率常数$1/3$都是尖锐的。
英文摘要
Let $λ_1(Ω)$ and $T(Ω)$ denote the first Dirichlet eigenvalue and torsional rigidity of a bounded convex domain $Ω\subset\mathbb{R}^2$, and let $M=\max_Ωu$, where $u$ is the torsion function. We prove the sharp inequalities $π^2/24<λ_1(Ω)T(Ω)/|Ω|<π^2/12$. This proves the two-dimensional case of Conjecture 4.2 proposed by van den Berg, Buttazzo, and Pratelli. Its planar formulation was later restated as Conjecture 1.1 by Bañuelos and Mariano, who proved it for triangles and rectangles. The lower bound follows by combining Payne's strict estimate for $λ_1M$ with the sharp torsion-efficiency inequality $T(Ω)\geq |Ω|M/3$. For smooth strictly convex domains, we use an Airy stress potential to construct a convex body whose surface-area measure is the torsional first-variation measure. A sharp one-dimensional inequality for directional maximum profiles then yields a stronger geometric containment. For the upper bound, we factor the Pólya functional through the second torsion moment. A level-set torsion--perimeter inequality gives the factor $5/6$, while a sharp weighted one-dimensional estimate gives the factor $π^2/10$. Collapsing triangles and elongating rectangles show that both endpoint constants, as well as the efficiency constant $1/3$, are sharp.
