  In this paper, we mainly study eigenvalue problems of p-Laplacian on domains
with an interior hole. Firstly we prove Faber-Krahn-type inequalities, and
Cheng-type eigenvalue comparison theorems on manifolds. Secondly, we prove a
comparison theorem for eigenvalues with inner Dirichlet and outer Neumann
boundary in minimal submanifolds in Euclidean space. Lastly we give a sharp
estimate of the eigenvalue (with outer Dirichlet and inner Neumann boundaries)
in terms of the Dircihlet eigenvalue, and also we give an upper bound of the
eigenvalue with inner Dirichlet and outer Neumann problems by the diameter of
the hole inside.
