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

Abstract

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