2023/11/21 by Chen, Hongbin, Gui, Changfeng, Yao, Ruofei · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2311.12659
This paper investigates the second Neumann eigenfunction u of a planar triangle T. In a recent paper by Judge and Mondal [Ann. Math., 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., 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. Amer. Math. Soc., 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.