关于多项式Faber-Krahn不等式
On the Polygonal Faber-Krahn Inequality
摘要：
Pólya和Szegö在70年前就证明了在具有$n$边和固定面积的多边形中，使Dirichlet-Laplace算子的第一特征值最小的平面集是正多边形。尽管这个结果看起来很简单，但它只在三角形和四边形中得到了证明。在本文中，我们证明了，对于每个$n \ge 5$的猜想的证明可以减少到有限数量的认证数值计算。此外，正多边形的局部极小值问题可以简化为一次数值计算。对于$n=5, 6,7, 8$，我们执行此计算，并通过有限元验证数值逼近，直至机器误差。
英文摘要：
It has been conjectured by Pólya and Szegö seventy years ago that the planar set which minimizes the first eigenvalue of the Dirichlet-Laplace operator among polygons with $n$ sides and fixed area is the regular polygon. Despite its apparent simplicity, this result has only been proved for triangles and quadrilaterals. In this paper we prove that for each $n \ge 5$ the proof of the conjecture can be reduced to a finite number of certified numerical computations. Moreover, the local minimality of the regular polygon can be reduced to a single numerical computation. For $n=5, 6,7, 8$ we perform this computation and certify the numerical approximation by finite elements, up to machine errors.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
