切向多边形的波利亚-塞格定理
A Quantitative Pólya--Szegő Theorem for Tangential Polygons
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中文总结 AI 辅助
该研究证明给定面积的切向N边形中，正N边形的扭转刚度最大，分解切向多边形为混合狄利克雷-纽曼直角三角形单元，推导了相关映射的严格凹性与亏空分解，还应用其证明等面积正多边形扭转刚度随边数增大并得到相关准则。
中文摘要 AI 辅助
我们证明，对于每个整数N≥3，在给定面积的所有切向N边形中，正N边形使扭转刚度达到唯一最大值。由于每个三角形都是切向的，N=3的情形给出了经典三角形波利亚-塞格定理的独立证明。该证明将切向多边形分解为混合狄利克雷-纽曼直角三角形单元，其分析核心是映射α→h(tanα)−1/8 tanα的严格凹性，其中h是具有一条狄利克雷边和两条纽曼边的直角三角形单元的混合扭转刚度，α∈(0,π/2)是两条纽曼边的夹角。我们还得到了显式亏空分解，将角不对称性与周长过剩分离开来。作为应用，我们给出了一个新颖的分析证明，即等面积正多边形的扭转刚度随边数严格增大，并推导了一个显式准则，确保切向多边形的第一狄利克雷本征值大于等面积正多边形的对应值。
英文摘要
For a bounded Lipschitz domain $Ω\subset\mathbb R^2$, let $T(Ω)=\int_Ωu_Ω\, dx$ denote its torsional rigidity, where $-Δu_Ω=1$ in $Ω$ and $u_Ω=0$ on $\partialΩ$. We prove a quantitative Pólya--Szegő inequality for tangential polygons. Let $N\ge3$, let $P$ be a tangential $N$-gon, set $A=|P|$, and let $R_N$ be the regular $N$-gon of area $A$. Writing $L(\cdot)$ for perimeter, we obtain the explicit deficit estimate \[ T(R_N)-T(P)\ge \frac{A^2}{8N\tan(π/N)} \left(1-\frac{L(R_N)^2}{L(P)^2}\right)^2.\]Thus, at fixed area, the torsional deficit is controlled from below purely by the perimeter ratio. In particular, the regular $N$-gon is the unique maximizer of torsional rigidity among tangential $N$-gons of prescribed area; for $N=3$ this gives the classical triangular Pólya--Szegő theorem with a quantitative estimate. The same perimeter estimate yields an explicit positive lower bound for $T(R_{N+1})-T(R_N)$ for equal-area regular polygons, and hence a rather short alternative proof of the strict monotonicity of torsional rigidity in $N$. Combined with the Kohler--Jobin inequality, it also gives an explicit sufficient condition for the polygonal Faber--Krahn inequality within the tangential class. Our full quantitative inequality is stronger: it contains, in addition, a nonnegative angular Jensen deficit, which yields quantitative angular stability away from degenerate configurations.
