Some observations on the first eigenvalue of the p-Laplacian and its connections with asymmetry

In this work, we present a lower bound for the first eigenvalue of the p-Laplacian on bounded domains in $mathbb{R}^2$. Let $lambda_1$ be the first eigenvalue and $lambda_1^*$ be the first eigenvalue for the ball of the same volume. Then we show that $lambda_1gelambda_1^*(1+Calpha(Omega)^{3})$, for some constant $C$, where $alpha$ is the asymmetry of the domain $Omega$. This provides a lower bound sharper than the bound in Faber-Krahn inequality