Introduction to variational methods

• Grant number: 24/14947-4
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: October 01, 2024
• End date: September 30, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Ederson Moreira dos Santos
• Grantee: Matheus Henrique Lima e Silva
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated research grant: 22/16407-1 - TESdE: Thematic on Equations and Systems of differential Equations, AP.TEM

Abstract

Variational methods are one of the main tools for dealing with problems in the nonlinear theory of differential equations. The main purpose of this project is to show how this is done. The main idea is the formulation of an equivalent variational problem. The variational problem consists of obtaining critical points for the functional associated to the differential equation. This idea appeared in the mid-19th century, explicitly with Dirichlet and Riemann, with the problem now known as Dirichlet problem for Laplace equation. Hilbert was the pioneer in using this method specifically to prove the existence of a solution to the Dirichlet problem. It is unquestionable that Hilbert's work contributed to the development of the Lebesgue integral, functional analysis and topology. Through this project, we study how these theories make it possible to provide a simple and elegant treatment of variational problems.