TOPICS IN ADVANCED ANALYSIS A

Degree course: 
Corso di Second cycle degree in MATHEMATICS
Academic year when starting the degree: 
2023/2024
Year: 
1
Academic year in which the course will be held: 
2023/2024
Course type: 
Compulsory subjects, characteristic of the class
Credits: 
8
Period: 
Second semester
Standard lectures hours: 
72
Detail of lecture’s hours: 
Lesson (48 hours), Exercise (24 hours)
Requirements: 

Core courses in Analysis.

Final Examination: 
Orale

Written and oral examination.

Assessment: 
Voto Finale

Achieve advanced tools in Nonlinear Analysis and PDE which are introductory to contemporary research.

Euler –Lagrange equations and solutions of partial differential equations via the Dirichlet principle of minimal Energy. Towards weak solutions. A few facts from Functional Analysis: weak derivatives and Sobolev spaces, embedding inequalities, the Rellich-Kondrachov theorem, extensions and traces. A direct method in the Calculus of Variations, minima of weakly lower semicontinuous functionals: applications to nonlinear Schroedinger’s equation. The Nehari manifold and ground states solutions, bootstrap argument in elliptic regularity theory. Introduction to topological methods in Nonlinear Analysis for indefinite functionals: deformation lemma and the mountain-pass theorem by Ambrosetti-Rabinowitz, applications to semilinear elliptic equations. The Ekeland Variational Principle. The effect of Symmetry, Critical growth problems, lack of compactness and Pohozaev identity. Quantization of energy and the Brezis-Nirenberg theorem.

Euler –Lagrange equations and solutions of partial differential equations via the Dirichlet principle of minimal Energy. Towards weak solutions. A few facts from Functional Analysis: weak derivatives and Sobolev spaces, embedding inequalities, the Rellich-Kondrachov theorem, extensions and traces. A direct method in the Calculus of Variations, minima of weakly lower semicontinuous functionals: applications to nonlinear Schroedinger’s equation. The Nehari manifold and ground states solutions, bootstrap argument in elliptic regularity theory. Introduction to topological methods in Nonlinear Analysis for indefinite functionals: deformation lemma and the mountain-pass theorem by Ambrosetti-Rabinowitz, applications to semilinear elliptic equations. The Ekeland Variational Principle. The effect of Symmetry, Critical growth problems, lack of compactness and Pohozaev identity. Quantization of energy and the Brezis-Nirenberg theorem.

Lectures.

This course aims at introducing students to contemporary research in Nonlinear Analysis and Nonlinear Partial Differential Equations.