Uniqueness of essential factors of the second Neumann eigenfunctions on triangles
Authors: Hongbin Chen, Changfeng Gui, Ruofei Yao
Summary: This paper offers with the second Neumann eigenfunction u of any planar triangle T. In a latest work by C. Choose and S. Mondal [Ann. Math., 2022], it was established that u doesn’t have any essential level inside the inside of T. On this paper, we present the distinctiveness of non-vertex essential level and the monotonicity property of the second eigenfunction. To be extra exact, when T will not be an equilateral triangle, the non-vertex essential level exists if and provided that T is an acute triangle that isn’t a super-equilateral triangle, and the worldwide extrema of u are achieved at and solely on the endpoints of the longest aspect. This establishes the origin theorem and conjecture 13.6 initially posed by C. Choose and S. Mondal [Ann. Math., 2020]. Our proof depends closely on continuity strategies, eigenvalue inequalities, and the utmost precept to ascertain these outcomes
