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

The course explores some variational aspects in the so-called Critical Point Theory.
In particular, the functionals considered are generally undefined, so the classical (or direct) methods of the Calculus of Variations do not apply.
The techniques used are therefore of a topological nature: minimax, mountain pass, linking theorems in general.
The tools used in these methods are essentially due to a suitable compactness property (Palais-Smale) and to the Deformation Lemma.
As a consequence of the Theory, existence theorems of solutions for differential equations are obtained, which correspond to the Euler-Lagrange equations of suitable functionals.
As applications, two types of problems are considered in particular, of a geometric/physical nature: in the first case, solutions of elliptic PDEs will be treated; in the second periodic orbits of Hamiltonian systems will be studied.
The course is essentially self-contained: the necessary notions will be recalled or defined in the preliminary phase, in any case the fundamental courses of Analysis of the Bachelor's Degree are assumed as prerequisites.

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

 

However, the course will be taught in Italian: see the italian version for the detailed program.

46146 - ANALISI NON LINEARE

Readings/Bibliography

- 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 in classroom.

Assessment methods

Final oral exam.

Teaching tools

Additional material can be found on Virtuale.

Office hours

See the website of Vittorio Martino