On the nodal lines of second eigenfunctions of the free membrane problem

Abstract

For certain doubly symmetric bounded domains in the plane we consider the eigenvalue problem for the Laplacian with Neumann boundary condition. Taking domains with the symmetry group of the square as a starting point, we show that unimodular linear transformations preserving the axes deform ω into domains with a simple second eigenvalue. We also show that the nodal sets of the second eigenfunctions lie on one of the axes

Keywords:

• Eigenvalues
• multiplicity
• eigenfunctions
• nodal sets

AMS(MOS)::

• 35B05
• 35p99