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96747 - Advanced Theory of Partial Differential Equations

Academic Year 2026/2027

                
                        Docente:
                        Nicola Abatangelo
                    
                        Credits:
                        6
                    
                        SSD:
                        MATH-03/A
                    
                        Language:
                        Italian
                    
                        Moduli:
                        
                            Nicola Abatangelo
                            (Modulo 1)
                        
                            Eugenio Vecchi
                            (Modulo 2)
                        
                        Teaching Mode:
                        
                                    In-person learning (entirely or partially) (Modulo 1); 
                                
                                    In-person learning (entirely or partially) (Modulo 2)
                                
                            Campus:
                            Bologna
                        
                            Corso:
                            Second cycle degree programme (LM) in
                            Mathematics (cod. 6730)

                            Teaching resources on Virtuale
                        
                                        Course Timetable
                                    
from Sep 15, 2026 to Dec 17, 2026

Learning outcomes

At the end of the course, the student possesses advanced knowledge of the theory of some relevant partial differential equations. In particular, he/she is able to obtain fine properties of the solutions of the equation and associated problems.

Course contents

The course is divided into two modules, both of which cover the theory of linear and semilinear elliptic equations.

The first part covers standard theory regarding the existence and uniqueness of solutions.

MODULE 1 (N. Abatangelo)

* harmonic functions, the fundamental solution of the Laplacian, and the Poisson kernel;
* subharmonic functions and Perron’s method for the existence of solutions to linear problems;
* weak and strong maximum principles, classical and on small domains, Hopf’s lemma;
* methods for the existence of solutions on bounded domains;
comparison principles and uniqueness of solutions.

The second part concerns the qualitative properties of solutions and non-existence results.

MODULE 2 (E. Vecchi)

* symmetry and monotonicity using the technique of moving planes and moving spheres;
* non-existence of solutions in various contexts: via moving planes in the entire space, via the Pohozaev identity in star-shaped domains;
* the overdetermined problem and the rigidity of balls: the approaches of Serrin and Weinberger;
* an overview of Alexandrov’s soap bubble theorem and Saint-Venant’s torsion problem.

Readings/Bibliography

S. Dipierro and E. Valdinoci, Elliptic partial differential equations from an elementary viewpoint---a fresh glance at the classical theory, World Scientific Publishing Co., 2024.
P. Quittner and Ph. Souplet, Superlinear parabolic problems, Birkhäuser, 2019.
A. Ambrosetti and D. Arcoya, An introduction to nonlinear functional analysis and elliptic problems, Birkhäuser, 2011.
A. Ambrosetti and A. Malchiodi, Nonlinear analysis and semilinear elliptic problems, Cambridge Univ. Press, 2007.

Teaching methods

Lectures at the blackboard.

Assessment methods

Choice of traditional oral examination or seminar on a topic related to the course contents.

Teaching tools

Essential notes uploaded on Virtuale.

Office hours

See the website of Nicola Abatangelo

See the website of Eugenio Vecchi

SDGs

This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.