97-417 Hoffmann-Ostenhof M., Hoffmann-Ostenhof T., Nadirashvili N.

On the Nodal Line Conjecture (522K, Postscript) Jul 24, 97

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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.

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