Eigenvalues Estimates for the p-Laplace Operator on Manifolds

The Laplace-Beltrami operator on a Riemannian manifold, its spectral theory and the relations between its first eigenvalue and the geometrical data of the manifold, such as curvatures, diameter, injectivity radius, volume, has been extensively studied in the recent mathematical literature. In the last few years, another operator, called p-Laplacian, arising from problems on Non-Newtonian Fluids, Glaceology, Nonlinear Elasticity, and in problems of Nonlinear Partial Differential Equations came to the light of Geometry. Since then, geometers showed that this operator exhibit some very interesting analogies with the Laplacian. Let (M,g) be a smooth Riemannian manifold and Ω ⊂ M a domain. For 1 < p < ∞, the p-laplacian on Ω is defined by △p(u) = −div [