Não existência de autovalores do operador de Laplace-Beltrami em gráficos radiais

Abstract

In this work we study the Laplace-Beltrami operator defined on Riemannian manifolds. In addition to the spectrum of such an operator, we also present some of its properties, such as the fact that this operator is self-adjoint and non-negative. Our main goal is to analyze the existence of eigenvalues for the Laplace-Beltrami operator, under certain conditions, for exemple, surfaces that are complete graphs of radial functions, which is a revolution non-compact surfaces. This dissertation is based on the article "On the spectrum of the Laplace-Beltrami Operator on the Non-Compact Surface"of Takao Tayoshi(Comm. By Kinjiro Kunugi, MJA, Feb. 12, 1971). To perform this work were introduced basics concepts of functional analysis, with emphasis on the study of Hilbert spaces and the spectral theory of self-adjoint operators, Riemannian Geometry in surfaces and Partial Differential Equations, in particular results for elliptic operators of second order.In addition, were needed some results for advanced mathematics.