An Extension of Barta's Theorem and Geometric Applications

We prove an extension of a theorem of Barta then we make few geometric applications. We extend Cheng's lower eigenvalue estimates of normal geodesic balls. We generalize Cheng-Li-Yau eigenvalue estimates of minimal submanifolds of the space forms. We prove an stability theorem for minimal hypersurfaces of the Euclidean space, giving a converse statement of a result of Schoen. Finally we prove a generalization of a result of Kazdan-Kramer about existence of solutions of certain quasi-linear elliptic equations.