Foundations of Computational Mathematics ( IF 3 ) Pub Date : 2020-02-19 , DOI: 10.1007/s10208-020-09447-y Nilima Nigam , Bartłomiej Siudeja , Benjamin Young
A modified version of Schiffer’s conjecture on a regular pentagon states that Neumann eigenfunctions of the Laplacian do not change sign on the boundary. In a companion paper by Bartłomiej Siudeja it was shown that eigenfunctions which are strictly positive on the boundary exist on regular polygons with at least 6 sides, while on equilateral triangles and cubes it is not even possible to find an eigenfunction which is nonnegative on the boundary. The case for the regular pentagon is more challenging, and has resisted a completely analytic attack. In this paper, we present a validated numerical method to prove this case, which involves iteratively bounding eigenvalues for a sequence of subdomains of the triangle. We use a learning algorithm to find and optimize this sequence of subdomains, making it straightforward to check our computations with standard software. Our proof has a short proof certificate, is checkable without specialized software and is adaptable to other situations.
中文翻译:
正则五边形上Schiffer猜想的有限元证明
Schiffer猜想在规则五边形上的修改版本指出,拉普拉斯算子的Neumann本征函数不会改变边界上的符号。在BartłomiejSiudeja的同伴论文中表明,在边界上严格为正的本征函数存在于至少具有6个边的规则多边形上,而在等边三角形和立方体上,甚至不可能找到在边界上为非负的本征函数。正五边形的情况更具挑战性,并且抵抗了完全的分析攻击。在本文中,我们提出了一种经过验证的数值方法来证明这种情况,该方法涉及迭代限制三角形子域序列的特征值。我们使用学习算法来查找和优化此子域序列,使用标准软件即可轻松检查我们的计算。我们的证明具有简短证明,可以在不使用专门软件的情况下进行检查,并适用于其他情况。