The Second Eigenvalue of the p-Laplacian as p Goes to 1

Abstract

The asymptotic behaviour of the second eigenvalue of the $p$-Laplacian operator as $p$ goes to 1 is investigated. The limit setting depends only on the geometry of the domain. In the particular case of a planar disc, it is possible to show that the second eigenfunctions are nonradial if $p$ is close enough to 1.