A ug 2 00 3 Dirichlet Eigenvalues Estimates and Quasilinear Elliptic Equations

First, we give an estimable lower bound expression in terms of certain vector fields for the fundamental tone of a bounded domain in a Riemannian manifold (Theorem 2.5). This result can be view as an extension of a theorem of Barta. Second, we relate this Barta's theorem to restrictions to the existence of solutions of certain quasilinear equations (Remark 2.2, Theorem 2.7 and Theorem 2.9). Third, this lower bound in terms of vector fields allow us to improve the well known Cheng's lower estimates beyond the cut locus, provided the cut locus has codimension at least two (Theorem 3.1, Theorem 3.3). Finally we make an application to stability of minimal hypersurfaces.