Aim to introduce the student to PDEs starting from basic notion to some of the most recent advances, in order to know some fundamental tools used in PDE research.
The course assumes some background in the theory of Banach spaces, Fourier transform, distributions, tempered distributions.
For some advanced material some additional tools in analysis will be presented. Interested students who don’t have such a mathematical background shouldn’t be discouraged to attend the course, because to get a sense of what will be taught in the course it is not necessary to have rigorous understanding but rather an intuitive grasp of such background that the lecturer can provide to these students upon request.
However, to fully benefit from this course, it is necessary that the students fill the background gaps.

Functional analysis, specifically Sobolev spaces and, broadly speaking, the topics of the 1st year courses ADVANCED ANALYSIS parts A e B.

The course is focused on Partial Differential Equations. It will start from classical theorems up to some nonlinear PDEs under active research.
1) Basic notions and definitions, Cauchy-Kowaleskaya Theorem
2) Some fundamental examples of linear PDEs: transport, Laplace, heat, wave equations
3) Some examples of fundamental nonlinear PDEs: Hamilton-Jacobi, conservation laws
4) Linear PDEs (elliptic, parabolic, wave): existence of solutions, regularity (with or without boundary)
5) Nonlinear PDEs and Calculus of Variation, Hamilton-Jacobi equations, Conservation Laws If time permits, the course will cover recent results for some nonlinear PDEs listed above.

1) L.C. Evans, Partial Differential Equations
2) Research papers on particular PDEs (transport, Conservation Laws, Hamilton-Jacobi), which will be given during the course

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The course consists of lectures during which the Instructor discusses all the details of the topics covered, answers student's questions and tries to get them involved. The students will receive before the lectures the lecture notes of the Instructor

The lecture notes and other information will be available through Moodle

The exam consists of a student seminar of about 30 minutes on a topic arranged with the Instructor, during which the student will show whether or not is able to apply the main ideas presented during the lectures by the Instructor in specific and analogous contexts. The Instructor might ask some questions on the topics covered during the course in class.