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On the nonexistence of some special eigenfunctions for the Dirichlet Laplacian and the Lamé system. (English) Zbl 0934.74010
Summary: Let $$\Omega$$ be a bounded Lipschitz domain in $$\mathbb{R}^n$$, $$n\geq 3$$. We prove that the Dirichlet Laplacian does not admit any eigenfunction of the form $$u(x)=\varphi(x')+\psi(x_n)$$ with $$x'=(x_1,\dots,x_{n-1})$$. The result is sharp since there are two-dimensional polygonal domains in which this kind of eigenfunction 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 three-dimensional case for which the Lamé system with Dirichlet boundary conditions admits uniaxial eigenvibrations.

##### MSC:
 74B05 Classical linear elasticity 35Q72 Other PDE from mechanics (MSC2000) 35P99 Spectral theory and eigenvalue problems for partial differential equations 74H45 Vibrations in dynamical problems in solid mechanics
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