Optimization of the first eigenvalue in problems involving the $p$-Laplacian

This paper concerns minimization and maximization of the first eigenvalue in problems involving the $p$-Laplacian, under homogeneous Dirichlet boundary conditions. Physically, in the case of $N=2$ and $p$ close to $2$, our equation models the vibration of a nonhomogeneous membrane $\Omega$ which is fixed along the boundary. Given several materials (with different densities) of total extension $\vert\Omega\vert$, we investigate the location of these material inside $\Omega$ so as to minimize or maximize the first mode in the vibration of the membrane.