This paper investigates the second Neumann eigenfunction \ (u\) of a planar triangle \ (T\). In a recent paper by Judge and Mondal (Ann. Math. (2) 195 (1): 337–362, 2022), it was shown that \ (u\) has no critical points in the interior of \ (T\). In this paper, we show that \ (u\) has at most one non-vertex critical point and that \ (u\) is monotone in a certain direction in \ (T\). More precisely, when \ (T\) is not equilateral, we show that \ (u\) vanishes at some vertex if and only if \ (T\) is superequilateral, and that \ (u\) has a non-vertex critical point if and only if \ (T\) is acute and not superequilateral. These results confirm both the original theorem and Conjecture 13. 6 of Judge and Mondal (Ann. Math. (2) 191 (1): 167–211, 2020). We also resolve the objective of Polymath 7 (research thread 1), namely, that the extrema of \ (u\) are attained only at the endpoints of the longest side. In addition, we settle a conjecture of Siudeja (Proc. Am. Math. Soc. 144 (6): 2479–2493, 2016) on the ordering of mixed Dirichlet–Neumann Laplacian eigenvalues for triangles. Our proofs combine the continuity method, eigenvalue inequalities, the maximum principle, and the moving plane method.
Kim et al. (Tue,) studied this question.