Abstract
Let Ω be a bounded Lipschitz domain in ℝn with n ≥ 3. We prove that the Dirichlet Laplacian does not admit any eigenfunction of the form u(x) =ϕ(x′)+ψ(x n) with x′=(x1, ..., x n−1). The result is sharp since there are 2-d polygonal domains in which this kind of eigenfunctions does exist. These special eigenfunctions for the Dirichlet Laplacian are related to the existence of uniaxial eigenvibrations for the Lamé system with Dirichlet boundary conditions. Thus, as a corollary of this result, we deduce that there is no bounded Lipschitz domain in 3-d for which the Lamé system with Dirichlet boundary conditions admits uniaxial eigenvibrations.
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Sweers, G., Zuazua, E. On the Nonexistence of Some Special Eigenfunctions for the Dirichlet Laplacian and the Lamé System. Journal of Elasticity 52, 111–120 (1998). https://doi.org/10.1023/A:1007524411396
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DOI: https://doi.org/10.1023/A:1007524411396