论文标题
在多边形的Faber-Krahn不平等
On the Polygonal Faber-Krahn Inequality
论文作者
论文摘要
Pólya和SzegöseveSevene曾经猜想，平面套件将Dirichlet-Laplace运算符的第一个特征值最小化，该多边形是$ n $侧面和固定面积的多边形。尽管显而易见，但该结果仅用于三角形和四边形。在本文中，我们证明，对于每个$ n \ ge 5 $，可以将猜想的证据简化为有限数量的认证数值计算。此外，常规多边形的局部最小值可以简化为单个数值计算。对于$ n = 5，6,7，8 $，我们执行此计算并通过有限元素来证明数值近似，直到机器错误。
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.
