Isoperimetric inequalities for the first eigenvalue of the p-Laplacian

Abstract

In this talk, we discuss some isoperimetric problems for the first eigenvalue of the p-Laplace operator. We start with classical isoperimetric inequalities with different boundary conditions. Then we move on to the analogous isoperimetric inequalities with the mixed Dirichlet-Neumann boundary conditions on multiply-connected domains in R

N (N ≥ 2). We establish that a suitable concentric annular domain produces (uniquely) the largest first eigenvalue among all multiply-connected Lipschitz domains of a given volume. Furthermore, we talk about some monotonicity results (with respect to domain perturbation) for the first Dirichlet eigenvalue of the p-Laplacian on a domain with certain dihedral symmetry. In addition, we introduce the optimization problems for the first weighted Dirichlet eigenvalue. We prove the existence of an optimal weight and study various symmetry results (Steiner symmetry, Foliated Schwarz symmetry, etc.) for the optimal weights and the associated eigenfunctions.