This learning unit (LU) aims at providing knowledge and competencies in some methods in nonlinear mathematical analysis, particularly in topological and variational methods, and their applications to the study of existence of solutions of differential equations.
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.
