The course aims to give a general overview of the theoretical aspects of the most important partial differential equations arising as fundamental models in the description of main phenomena in Physics, Biology, economical/social sciences and data analysis, such as diffusion, transport, reaction, concentration, wave propagation, with a particular focus on well-posedness (i.e. existence, uniqueness, stability with respect to data). Moreover, the theoretical properties of solutions are studied in connection with numerical approximation methods (e.g. Galerkin finite dimensional approximations) which are studied and implemented in the Numerical Analysis courses.
Derivation of some partial differential equations from the modelling.
Partial differential equations of first order: characteristics' method, eikonal equation. Weak solutions: scalar Conservation Law, introduction to the Calculus of Variations and to the Hamilton-Jacobi equation.
Linear partial differential equations of second order: classification.
Laplace equation and Poisson equation: fundamental solution, harmonic functions, Green's identity, Green's function, Poisson's formula for the ball, gradient estimates, Liouville's Theorem.
Elliptic equations: maximum principles, Hopf Lemma. Uniqueness theorems. Existence theorems: weak solutions via Lax-Milgram Theorem and classical solutions via Perron's method.
Introduction to the heat equation and to the wave equation.
Parabolic and hyperbolic equations: Galerkin method, introduction to Semigroup Theory.
|Yehuda Pinchover, Jacob Rubinstein||An Introduction to Partial Differential Equations||Cambridge||2005|
|Qing Han, Fanghua Lin||Elliptic Partial Differential Equations||American Mathematical Society||2011|
|D. Gilbarg - N. S. Trudinger||Elliptic Partial Differential Equations of Second Order||Springer||1998||3-540-13025-X|
|Evans, L. C.||Partial Differential Equations (Edizione 1)||American Mathematical Society||1998||0821807722|
|András Vasy||Partial Differential Equations - An Accessible Route through Theory and Applications||American Mathematical Society||2015||978-1-4704-1881-6|
|S. Salsa||Partial Differential Equations in Action||Springer Verlag Italia||2008||978-88-470-0751-2|
REGULATIONS for the summer exam session in PDE. - COVID19 EMERGENCY
1. Oral exame with remote connection, dates to be arranged with the teachers.
2. Everybody must communicate his/her availability for the exam BEFORE 15th June, in order to compile a schedule in a reasonable amount of time.
3. Max 2 attempts. If you fail two times, your next attempt will be in the session of September. No exceptions.
4. Contents of the exam: the exam will concern for 2/3 on the first part, and for 1/3 on the second part, roughly according the partition of the lecture hours.
5. Description of the exam: in the exam there will be
- a practical part (which will play the role of the written part) in which it will be requested the solution of an exercise on a topic among: method of characteristics, conservation laws and Riemann's problem, calculus of variations.
- a theoretical question on the fist part of the course.
- a theoretical question on the second part of the course.
6. The knowledge of all the definition and statements is mandatory. At the beginning of June we will publish a list of the proofs that will be requested to know (we will not publish it before the beginnning of June for reasons that should be clear).
OLD REGULATIONS (SUSPENDED)
The exam will consist in an oral examination based on all the topics covered by the lectures. More precisely, the examination will be made of 3 steps: in the first step the student will be asked to report in details on a result randomly selected from a previously arranged list of results. Only if the student passes the first step he/she will be admitted to the second step of the examination, which will consist in a more general discussion on themes of the programme. In the third and last step the student will be asked to present his/her favourite topic.
The assessment will be based on the extent to which the student will overlook and master the main ideas and mathematical tools/techniques delivered by the lecturer during the course.