Introdução aos Métodos Variacionais

Abstract

This monograph presents an introduction to Variational Methods, which today form an important method that is applied in the field of differential equations. Thus, we sought to answer the following guiding question: how to solve ordinary differential equations (ode) using Variational Methods? This research has as main objective to determine necessary and sufficient conditions for certain ordinary differential equations to have a solution via Variational Methods. For this, initially, a brief review of measure and Lebesgue spaces was made to generalize the concept of Riemann’s integral; from this, the concept of the weak derivative was defined, followed by the well-known Sobolev spaces. Thus, in these spaces, what we call a weak solution of the functional associated with the given equation was established, to later solve the Edo by the Variational Methods. As for the methodology used in this work, we have exploratory and bibliographical research, and a qualitative approach. As a result of this study, we highlight the use of the Mountain Pass Theorem, which provides some functional conditions, including the Palais Smale condition, under which the functional has a critical point. Thus, Variational Methods are concerned with finding critical functional points associated with some differential equation.