Visitando algumas ferramentas da análise moderna para soluções EDP's

Abstract

In 1856, Riemann introduced the Dirichlet Principle as an innovative approach to the Dirichlet Problem, proposing a treatment based on a variational problem. Essentially, this principle aimed to minimize an energy functional within a class of admissible functions. Despite its imprecisions, the Dirichlet Principle was adopted by renowned mathematicians, including Riemann himself, as evidenced in its application in the proof of the Conformal Mapping Theorem. However, around 1870, the mathematical community underwent revisions that questioned previously considered evident concepts. In this context, examples contradicting the Dirichlet Principle emerged, necessitating a correction. The solution came with the theory of Sobolev spaces, developed by Sergei Sobolev. These spaces provide an elegant mathematical framework, extending the concept of derivatives beyond smooth functions, allowing the analysis of functions with weak or even distributional derivatives. Sobolev spaces, in turn, play a fundamental role in mathematical analysis, finding applicati- ons in various areas, from the theory of partial differential equations to optimization and differential geometry. This work aims to delve deeply into this theory, relying on references such as (PONCE, 2009), (EVANS, 2010), (FOLLAND, 1999), (BARTLE, 1995), (BREZIS, 2010), (WILLEM, 2013), and (STRUWE, 2008). In this research, the focus shifts to the resolution of Partial Differential Equations, highligh- ting the Lax-Milgram Theorem, Fredholm’s Alternative, and the application of Sobolev spaces in the analysis of a specific problem. The ultimate goal is to acquire a comprehensive and solid understanding, enabling the attainment of robust analytical results to address practical challenges in solving problems involving partial differential equations.