Generalization of Philippin's results for the first Robin eigenvalue and estimates for eigenvalues of the bi-drifting Laplacian

摘要

In the present paper, we first consider the weighted eigenvalue problem fu=fu in M with the Robin boundary condition +u=0 on M, where (Mn,g,e-f) is a compact n-dimensional weighted Riemannian manifold of nonnegative Bakry-Emery Ricci curvature. We derive under some convexity condition of the boundary M, an explicit lower bound of the first weighted Robin eigenvalue 1,f() depending only on the geometry of M and the constant appearing in the boundary condition. Another new estimate for 1,f() with respect to the first nonzero Neumann eigenvalue 2,f of the weighted Laplacian f is also obtained. Furthermore, we provide some lower bounds for the first buckling and clamped plate eigenvalues of the bi-drifting Laplacian on weighted manifolds.