Some Observations on the First Eigenvalue of the p-Laplacian and its Connections with Asymmetry

Abstract

In this work, we present a lower bound for the first eigenvalue of the p-Laplacian on bounded domains in ℝ2. Let λ1 be the first eigenvalue and λ*1 be the first eigenvalue for the ball of the same volume. Then we show that, λ1 ≥ λ*1 (1 + Cα(Ω)3, for some constant C, where α is the asymmetry of the domain Ω. This provides a lower bound sharper than the bound in Faber-Krahn inequality.