Eigenvalues of the Neumann Laplacian in symmetric regions

Abstract

In this work we are concerned with the multiplicity of the eigenvalues of the Neumann Laplacian in regions of Rn which are invariant under the natural action of a compact subgroup G of O(n). We give a partial positive answer (in the Neumann case) to a conjecture of V. Arnold [1] on the transversality of the transformation given by the Dirichlet integral to the stratification in the space of quadratic forms according to the multiplicities of the eigenvalues. We show, for some classes of subgroups of O(n) that, generically in the set of G-invariant, C2-regions, the action is irreducible in each eigenspace. These classes include finite subgroups with irreducible representations of dimension not greater than 2 and, in the case n=2, any compact subgroup of O(2). We also obtain some partial results for general compact subgroups of O(n).