- Docente: Vittorio Martino
- Credits: 6
- SSD: MAT/05
- Language: Italian
- Moduli: Giovanna Citti (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. 5827)
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from Sep 22, 2022 to Oct 13, 2022
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from Oct 18, 2022 to Dec 20, 2022
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