Initiation to Variational Inequalities

• Grant number: 24/15858-5
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: December 01, 2024
• End date: July 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Olimpio Hiroshi Miyagaki
• Grantee: Luana Moura de Oliveira
• Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil
• Associated research grant: 22/16407-1 - TESdE: Thematic on Equations and Systems of differential Equations, AP.TEM

Abstract

The main objective of this project is to study introductory topics in Variational Inequalities. In this regard, the initial focus of this work will be on the study of Green's Theorem and Gauss's Theorem, examining their implications for the calculation of line and surface integrals. Additionally, ordinary and partial differential equations will be explored, applying the mentioned theorems to solve practical problems related to the circulation and flux of vector fields in both planar and three-dimensional regions. Consequently, topics such as variable transformations in multiple integrals, second-order ordinary differential equations, and Fourier series, which are essential elements for the formulation and resolution of complex problems in physics and engineering, will also be considered as part of the foundational knowledge. The theoretical foundation will largely focus on the study of Variational Inequalities and their practical applications in relevant contexts, with an emphasis on the Obstacle Problem and the Rectangular Dikes Problem, a theory that will be the central study of this project. It is worth noting that the undergraduate mathematics curriculum briefly covers the subjects related topartial differential equations. Thus, this project also aims to present the main problems in the field and prepare the student to pursue potential postgraduate studies in this research area.