A Numerical Investigation of the Rayleigh$-$Faber$-$Krahn Inequality for Polyhedral Domains
We present numerical results for the problem of minimizing the first Dirichlet eigenvalue of the Laplacian among three-dimensional convex polyhedra with a prescribed number of facets. This problem can be viewed as a polyhedral version of the classical Rayleigh$-$Faber$-$Krahn inequality. Using a parameterization of polytopes by supporting hyperplanes, the problem is first reformulated as an equivalent finite-dimensional constrained minimization problem. A Lagrangian framework is employed for the numerical solution. Under the assumption that each vertex is incident to exactly $d$ facets, the sensitivity analysis of the Lagrangian combines finite-dimensional perturbations of the facets of $d$-dimensional polyhedra with infinite-dimensional shape derivatives of both the eigenvalue and the volume. Depending on the number of facets, the numerical results yield well-known polyhedra, such as Platonic solids, prisms, and the tetrakaidecahedron, or reveal nonregular optimal shapes exhibiting several symmetries.