On the lower bound of the principal eigenvalue of a nonlinear operator (Q2667168)

The author proves a sharp lower bound \(\lambda \geq \lambda_D\) for the first positive eigenvalue \(\lambda\) of a nonlinear elliptic diffusion operator of the \(p\)-Laplacian type satisfying the Bakry-Émery condition \(BE(\kappa,N)\) with \(\kappa \neq 0\), on a compact connected manifold \(M\), and the Neumann boundary conditions in the case \(\partial M \neq \emptyset\). Here, \(\lambda_D\) is the first positive Neumann eigenvalue of the corresponding one-dimensional problem. The proof is based on a gradient comparison technique together with maximum and diameter comparisons.