Estimativas para o primeiro autovalor positivo de um operador elíptico de segunda ordem na forma divergente e alguns teoremas de comparação

Abstract

In this thesis, we obtain lower bound estimates for the first eigenvalue of a second-order elliptic differential operator in divergence form on closed or compact with boundary weighted Riemannian manifolds. This operator generalizes operators such as the Laplacian, the drifted laplacian and the square of Cheng-Yau. For this, we use a known Bochner type formula for this operator, and a Reilly type formula obtained in this thesis. We also derive comparison results for the mean curvature of geodesic spheres, generalizing the local Laplacian comparison theorem.