  Let $(M^{n+1}, g)$ be a compact Riemannian manifold with smooth boundary B
and nonnegative Bakry-Emery Ricci curvature. In this paper, we use the
solvability of some elliptic equations to prove some estimates of the weighted
mean curvature and some related rigidity theorems. As their applications, we
obtain some lower bound estimate of the first nonzero eigenvalue of the
drifting Laplacian acting on functions on B and some corresponding rigidity
theorems.
