46146 - Nonlinear Analysis

Learning outcomes

At the end of the course the student knows some aspects of the theory of non linear systems with particular emphasis on PDEs and is able to recognize the principal peculiarities of nonlinearity and similarities or difference with linear analysis.

Course contents

Module 1
At the end of this module, the student knows Schauder fixed point theory and applications to non linear PDE.

Program:
- Outline of degree theory
- Schauder fixed point theorem
- Leray Schauder theorem
- Application to solution of quasilinear elliptic equations with Schauder method of a priori estimates



Module 2
At the end of this module, the student knows the basic ideas and tecniques on minimax methods in the variational theory of critical points.

Program:
- Palais-Smale compactness condition
- Deformation lemma
- Mountain pass theorem
- Applications to elliptic PDEs
- Minimax principle
- Properties of linking method
- Applications to Hamiltonian systems

Readings/Bibliography

Module 1:
- K. Deimling, Nonlinear Functional Analysis
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order


Module 2:
- M.Struwe, Variational Methods; Springer
- A.Ambrosetti, A.Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems; Cambridge University Press
- P.H.Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations; AMS-CBMS

Teaching methods

Lectures.

Assessment methods

Final oral exam.

Teaching tools

Useful material for the course will be posted on Virtuale

Office hours

See the website of Vittorio Martino

See the website of Giovanna Citti