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In this paper, we obtain lower bounds for the first eigenvalue to some kindsof the eigenvalue problems for Bi-drifted Laplacian operator on compactmanifolds (also called a smooth metric measure space) with boundary and$m$-Bakry-Emery Ricci curvature or Bakry-Emery Ricci curvature bounded below.We also address the eigenvalue problem with Wentzell-type boundary conditionfor drifted Laplacian on smooth metric measure space.

Estimates for the first eigenvalues of Bi-drifted Laplacian on smooth metric measure space