Comment. Math. Helvetici 65 (1990) 96-103 0010-2571/90/010096-0851.50 + 0.20/0 9 1990 Birkh~iuserVerlag, Basel On the nodal lines of second eigenfunctions of the fixed membrane problem ROLF POTTER Abstract. A well-known conjecture about the second eigenfunction of a bounded domain in Rz states that the nodal line has to intersect the boundary in exactly two points. We give sufficient conditions on the domain for this assertion to hold. For special doubly symmetric domains we also prove that 22 is simple and that the nodal line of the second eigenfunction lies on one of the axes. 1. Introduction Consider the Dirichlet eigenvalue problem for the Laplacian on a bounded domain {2 c R 2 with boundary of class C2"~: Au+2u=O in t~ u = 0 on Off. (1.1) The set of eigenvalues can be arranged in a nondecreasing sequence of positive numbers tending to infinity 0 < 21 < 22 ~ ha " ' " 9 The corresponding eigenfunctions {//i }T=1 are in C2'~(t~ -) (see [3], Theorem 6.15) and analytic in the interior of t2. If u is an eigenfunction N(u) := {x e t2 : u(x) = 0} is called the nodal set of u; the connected components of t2\N(u) are called nodal domains. The Courant nodal domain theorem states that the i-th eigenfunction can possess at most i nodal domains. As a consequence of Courant's theorem, u~ has exactly one and //2 has exactly two nodal domains. (1.2) Cheng proved in [1] that, for any eigenfunction u, the nodal set N(u) consists of a finite number of Cl-immersed arcs ~ : (0, l) -~t2 or circles @ : S 1 -~fl. When these arcs or circles intersect or self-intersect, they form an equiangular system. As a consequence of (1.2), we have: If t2 is simply connected, N(u2) consists of one embedded arc or one embedded circle only. (1.3) 96