Hoffmann-Ostenhof M., Hoffmann-Ostenhof T., Nadirashvili N.
On the Nodal Line Conjecture
(522K, Postscript)
ABSTRACT. We consider Dirichlet eigenfunctions of membrane problems. A counterexample
to Payne's nodal line conjecture is given, i.e. a domain in $\mathbb R^2$
(not simply connected) whose second eigenfunction has a nodal set disjoint
from the boundary. Also a domain in $\mathbb R^2$ is given whose second
eigenvalue has multiplicity three.
Furthermore, some sufficient conditions are given which imply that an
eigenfunction of a Dirichlet membrane problem in $\mathbb R^n$ has a zero
set which hits the boundary.