论文标题
三角形和四边形的基本比率的全球和局部界限
Global and Local Bounds on the Fundamental Ratio of Triangles and Quadrilaterals
论文作者
论文摘要
我们提出了一个新的，计算机辅助的证据，证明了平面中的所有三角形，等边三角形独特地最大化了前两个Dirichlet-Laplacian特征值的比率。这证明了独立的证据，三角形Ashbaugh-benguria-payne-polya-weinberger猜想首先在Arxiv中证明：0707.3631 [Math.sp]和Arxiv：2009.00927 [Math.sp]。受ARXIV的启发：1109.4117 [Math.sp]，主要方法是使用扰动估计来确定局部最佳选择，然后对基本比率进行连续性估算，以对参数空间进行严格的计算搜索。我们重复此证明的一部分，以表明正方形是飞机上四边形之间基本比的严格局部优化器
We present a new, computer-assisted, proof that for all triangles in the plane, the equilateral triangle uniquely maximizes the ratio of the first two Dirichlet-Laplacian eigenvalues. This proves an independent proof the triangular Ashbaugh-Benguria-Payne-Polya-Weinberger conjecture first proved in arXiv:0707.3631 [math.SP] and arXiv:2009.00927 [math.SP]. Inspired by arXiv:1109.4117 [math.SP], the primary method is to use a perturbative estimate to determine a local optimum, and to then use a continuity estimate for the fundamental ratio to perform a rigorous computational search of parameter space. We repeat a portion of this proof to show that the square is a strict local optimizer of the fundamental ratio among quadrilaterals in the plane
