Estimates for the First Nonzero Eigenvalue of Elliptic Operators in Divergence Form

We consider elliptic operators in divergence form L = div (Φ · grad) either on a closed Riemannian manifold or in a domain with compact closure and piecewise smooth boundary M where Φ: M → End(TM) is a positive definite symmetric smooth section of the bundle of all endomorphisms of TM. We show that the first nonzero L-eigenvalues in the closed or Dirichlet eigenvalue problems can be bounded in terms of the Laplacian eigenvalues in the respective eigenvalue problem and the eigenvalues of Φ. We also present a method to obtain lower bounds for first Dirichlet L-eigenvalue in terms of vector fields generalizing the main result of [5]. We apply these results to give lower bounds for the first eigenvalue of the Lr operators on hypersurfaces with locally bounded (r + 1)-mean curvature.