Abstract. In this note, we consider a curved tube with varying crosssection formed by rotating open bounded Euclidean domains with respect to a reference curve, and successfully give a lower bound to the threshold of the Laplacian on the tube, subject to Dirichlet boundary conditions on the surface and Neumann conditions at the ends of the tube. This generalizes the corresponding result in [1].

In this paper, we investigate the buckling problem of the drifting Laplacian of arbitrary order on a bounded connected domain in complete smooth metric measure spaces (SMMSs) supporting a special function, and successfully obtain a general inequality for its eigenvalues. By applying this general inequality, if the complete SMMSs considered satisfy some curvature constraints, we can obtain a universal inequalities for eigenvalues of this buckling problem.

In this paper, we study the eigenvalue problem of poly-drifting Laplacian on complete smooth metric measure space ( M , ⟨ , ⟩ , e − ϕ d v ) \left(M,\langle ,\rangle ,{e}^{-\phi }{\rm{d}}v) , with nonnegative weighted Ricci curvature Ric ϕ ≥ 0 {{\rm{Ric}}}^{\phi }\ge 0 for some ϕ ∈ C 2 ( M ) \phi \in {C}^{2}\left(M) , which is uniformly bounded from above, and successfully obtain several universal inequalities of this eigenvalue problem.

No abstract