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|
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.