Extremal basic frequency of non-homogeneous plates

In this paper we propose two numerical algorithms to derive the extremal principal eigenvalue of the bi-Laplacian operator under Navier boundary conditions or Dirichlet boundary conditions. Consider a non-homogeneous hinged or clamped plate $\mathrm{\Omega }$, the algorithms converge to the density functions on $\mathrm{\Omega }$ which they yield the maximum or minimum basic frequency of the plate.