96747 - Advanced Theory of Partial Derivate Equations

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 consists of two modules, both dealing with the theory of semilinear elliptic equations.

MODULE 2 (E. Vecchi)

• Weak and strong maximum principles, classical and on small domains; Hopf's lemma;

• Qualitative properties of classical solutions: symmetry and monotonicity via the moving plane technique;

• Overdetermined problem and ball rigidity: Serrin's and Weinberger's approaches;

• Hints on the Alexandrov's soap bubble theorem and the Saint-Venant's torsion problem.

MODULE 1 (N. Abatangelo)

• Methods for the existence of solutions on bounded domains;

• Comparison principles and uniqueness of the solution;

• Non-existence of solutions in various contexts: via moving planes in the full space, via the Pohozaev's identity in star-shaped domains, via Green's functions in half-spaces.

• Hints on a priori estimates and concavity of solutions.

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