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Journal of the London Mathematical Society Advance Access originally published online on February 21, 2008
Journal of the London Mathematical Society 2008 77(2):424-442; doi:10.1112/jlms/jdm122
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© 2008 London Mathematical Society

An eigenvalue problem for the biharmonic operator on Z2-symmetric regions

Antônio L. Pereira

Instituto de Matemática e Estatística
Universidade de São Paulo
Rua do Matão, 1010
Cidade Universitária
CEP 05508-090 São Paulo, SP
Brazil
alpereir@ime.usp.br

Marcone C. Pereira

Escola de Artes
Ciências e Humanidades
Universidade de São Paulo
Av. Arlindo Bettio, 100
Ermelino Matarazzo
CEP 03028-000 São Paulo, SP
Brazil

In this work we show that the eigenvalues of the Dirichlet problem for the biharmonic operator are generically simple in the set of Z2-symmetric regions of Rn, n≥2, with a suitable topology. To accomplish this, we combine Baire's lemma, a generalised version of the transversality theorem, due to Henry [Perturbation of the boundary in boundary value problems of PDEs, London Mathematical Society Lecture Note Series 318 (Cambridge University Press, 2005)], and the method of rapidly oscillating functions developed in [A. L. Pereira and M. C. Pereira, Mat. Contemp. 27 (2004) 225–241].

2000 Mathematics Subject Classification 35J40, 35B30, 58C40.

The first author is partially supported by CNPq-Brazil and Fapesp grants 2003/11021-7 03/10042-0. The second author is partially supported by Fapesp 2006/06278-7.

Received March 22, 2007; revised November 1, 2007; published online February 21, 2008.


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