Eigenvalue Estimate for a Weighted $\boldsymbol{p}$-Laplacian on Compact Manifolds with Boundary

Let $(M^n,g)$ be a compact Riemannian manifold with convex boundary, let $d\mu=e^{h(x)}\,dV(x)$ be a weighted measure on $M$, and let $\Delta_{\mu,p}$ be the corresponding weighted $p$-Laplacian on $M$. We obtain a lower bound for the first nonzero Neumann eigenvalue of $\Delta_{\mu,p}$.