Nodal and other properties of the second eigenfunction of the Laplacian in the plane

Abstract

Rules are established for the intersection of nodals at a boundary in the plane relevant to the second eigenfunction of the Laplacian. Employing these results together with regularity theorems related to odd reflection of solutions of the Helmholtz equation, as well as a variation of C.S. Lin’s analysis, the following theorem is revisited: The nodal curve of the second eigenstate of the Laplacian for bounded convex domains in the plane, with Dirichlet boundary conditions, is a simple curve that intersects the boundary in two distinct points. Application is made to the regular convex polygons with C n , n ≥ 2 C_{n}, n \geq 2 , symmetry and to convex billiards with smooth boundaries.