1 the target eigenvalue problem

$$\Omega:=(0,1)\times(0,1),$$

$$\Laplacian^2 u=\lambda u$$   in $\Omega$ $$,$$

$$\mu\Laplacian u+\left(1-\mu\right)\frac{\rd^2 u}{\rd n^2}=0$$   (on $\rd\Omega$ )$$,$$

$$\frac{\rd}{\rd n}\left(\Laplacian u+\left(1-\mu\right)\frac{\rd^2 u}{\rd \tau^2}\right)=0 \quad\text{(on $\rd\Omega$)},$$

$\mu$ : given positive constant (Poisson ratio of the material), $n$ : the outward unit normal vector of the boundary, $\tau$ : the unit tangential vector of the boundary.

$$0=\lambda_1=\lambda_2=\lambda_3<\lambda_4<\cdots$$