Bifurcation of minimal surfaces and the first eigenvalue of the Laplacian
| Full text | |
| Author(s): |
Total Authors: 2
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| Affiliation: | [1] Univ Fed Amazonas, ICE, Dept Matemat, BR-69077070 Manaus, AM - Brazil
[2] Univ Sao Paulo, Inst Matemat & Estat, BR-05508090 Sao Paulo, SP - Brazil
Total Affiliations: 2
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| Document type: | Journal article |
| Source: | Journal of Mathematical Physics; v. 56, n. 11 NOV 2015. |
| Web of Science Citations: | 1 |
| Abstract | |
In this work, we are concerned with the multiplicity of the eigenvalues of the Neumann Laplacian in regions of R-n 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 Arnol'd {[}Funct. Anal. Appl. 6, 94-101 (1972)] 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, C-2-regions, the action is irreducible in each eigenspace Ker(Delta + lambda). 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). (C) 2015 AIP Publishing LLC. (AU) | |