On the lower bound of the principal eigenvalue of a nonlinear operator

Abstract

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 ne 0. Our results extends the work of Koerber Valtorta (Calc Vari Partial Differ Equ. 57(2), 49 2018) for case kappa =0 and Naber–Valtorta (Math Z 277(3–4):867–891, 2014) for the p-Laplacian.