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Optimization of the first eigenvalue in problems involving the bi-Laplacian. (English) Zbl 1181.35154
Summary: This paper concerns minimization and maximization of the first eigenvalue in problems involving the bi-Laplacian under Dirichlet boundary conditions. Physically, in case of $N = 2$, our equation models the vibration of a non-homogeneous plate $\Omega $ which is clamped along the boundary. Given several materials (with different densities) of total extension $|\Omega |$, we investigate the location of these materials throughout $\Omega $ so to minimize or maximize the first eigenvalue in the vibration of the clamped plate.

35P15Estimation of eigenvalues and upper and lower bounds for PD operators
47A75Eigenvalue problems (linear operators)
74K20Plates (solid mechanics)
74P10Optimization of other properties (solid mechanics)
49K20Optimal control problems with PDE (optimality conditions)
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