Functional Analysis
Topological Methods in Analysis
Variational Methods

Upon conclusion of this LU the student should:
- know the topological and variational methods that were studied and their application to the study of nonlinear differential equations;
- have acquired enough familiarity with the arguments and techniques used in the proofs of the results studied, so that, afterwards, he/she can both apply them to different contexts, and proceed to produce original research studies in these topics.
 

The syllabus of this LU consists of the following points:
- Introduction: existence of solutions to differential equations and extrema of functionals in Hilbert spaces.
- Topological methods: fixed point theorems, topological degree, and their applications to nonlinear differential equations.
- Variational methods: critical points of functionals in linear spaces; critical points of functionals restricted to manifolds; saddle points, min-max methods. Applications.
 

- Ambrosetti & Malchiodi: Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge studies in advanced mathematics vol. 104, Cambridge University Press, Cambridge, 2007;
- Ciarlet: Linear and Nonlinear Functional Analysis with Applications, Society for Industrial and Applied Mathematics, Philadelphia, 2013;
- Schechter: An Introduction to Nonlinear Analysis, Cambridge studies in advanced mathematics vol. 95, Cambridge University Press, Cambridge, 2012.
 

Evaluation is made on individual basis and it involves the coexistence of two modes: continuous assessment (60%) and final evaluation (40%). Further information is detailed in the Learning Agreement of the course unit.