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 Lp on a smooth metric measure space, without boundary or with a convex boundary and Neumann boundary condition, satisfying BE(κ, N ) for some negative κ. Our results extends the work of Koerber[12] for case κ = 0 and Naber-Valtorta[10] for the p-Laplacian.