A Real Analyticity Result for Symmetric Functions of the Eigenvalues of a Domain-Dependent Neumann Problem for the Laplace Operator (original) (raw)
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Zeitschrift Fur Analysis Und Ihre Anwendungen, 2005
Let Ω be an open connected subset of R n for which the Poincaré inequality holds. We consider the Dirichlet eigenvalue problem for the Laplace operator in the open subset φ(Ω) of R n , where φ is a locally Lipschitz continuous homeomorphism of Ω onto φ(Ω). Then we show Lipschitz type inequalities for the reciprocals of the eigenvalues of the Rayleigh quotient φ(Ω) |Dv| 2 dy φ(Ω) |v| 2 dy upon variation of φ, which in particular yield inequalities for the proper eigenvalues of the Dirichlet Laplacian when we further assume that the imbedding of the Sobolev space W 1,2 0 (Ω) into the space L 2 (Ω) is compact. In this case, we prove the same type of inequalities for the projections onto the eigenspaces upon variation of φ.
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