On the Lower Bound of the Principal Eigenvalue of a Nonlinear Operator

We prove sharp lower bound estimates for the first nonzero eigenvalue of the non-linear elliptic diffusion operator $L_p$ on a smooth metric measure space, without boundary or with a convex boundary and Neumann boundary condition, satisfying $BE(\kappa ,N)$ for $\kappa \mathrm{\ne }0$. Our results extends the work of Koerber[5] for case $\kappa =0$ and Naber-Valtorta[10] for the $p$-Laplacian.