46146 - Nonlinear Analysis

Academic Year 2019/2020

  • Moduli: Simonetta Abenda (Modulo 1) Vittorio Martino (Modulo 2)
  • Teaching Mode: Traditional lectures (Modulo 1) Traditional lectures (Modulo 2)
  • Campus: Bologna
  • Corso: Second cycle degree programme (LM) in Mathematics (cod. 8208)

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 consists of two parts which will be taught in series in a self-consistent manner. No preliminary knowledge in the subject is needed except for the content of the main courses already present in the degree in Mathematics.

Module 1 (prof.ssa Simonetta Abenda): Introduction to intagrable Hamiltonian systems

At the end of this module, the student knows  basic ideas and techniques of the theory of integrable Hamiltonian systems. Program:

- Recap on the theory of finite dimensional Hamiltonian systems, symplectic varieties and Poisson brackets

- Completely integrable systems (Arnold-Liouville theorem), separation of variables and action-angle variables

- Algebraic integrability, Lax formalism, R-matrices, classical Yang-Baxter equation;

- Brief introduction to analytic theory on Riemann surfaces;

- Applications: Toda lattice and Korteweg- de Vries hierarchy

 

Module 2 (prof. Vittorio Martino): Introduction to the minimax methods in the variational theory of critical points:

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:

- V. I. Arnold, A.B. Givental: Symplectic Geometry, in Dynamical Systems IV, Encyclopedia of Mathematical Sciences, volume 4, V. I. Arnold and S.P. Novikov Editors, Springer (Capitolo 3)

O. Babelon, D. Bernard, M. Talon: Introduction to classical integrable systems, Cambridge Monographs on Mathematical Physics, Cambridge University Press (2003)

- M.A. Olshanetsky, A.M. Perelomov: Integrable systems and finite dimensional Lie algebras, Chapter1 of Integrable Systems II, in Dynamical Systems VII, Encyclopedia of Mathematical Sciences, volume 4, V. I. Arnold and S.P. Novikov Editors, Springer (Toda)

- B.A. Dubrovin: Integrable Systems and Riemann Surfaces
Lecture Notes (preliminary version), 2009

 

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

 

Traditional lectures

Assessment methods

 

Oral examination on the program of the course

Teaching tools

Module 1:

Useful material for this part of the course, like lecture notes, will be published on the electronic platform of the course.

 

Module 2:

The material on this part of the course will be published on the following web page: http://www.dm.unibo.it/~martino/teaching.html

Office hours

See the website of Simonetta Abenda

See the website of Vittorio Martino