Scientific Disciplinary Sector (SSD)
MAT/05 - MATHEMATICAL ANALYSIS
Language of instruction
II semestre dal Mar 1, 2021 al Jun 11, 2021.
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.
Generalities on PDEs, classification, examples.
First-order equations: the method of characteristics, examples.
Second-order equations: elliptic, parabolic, hyperbolic equations.
Laplace's equation: physical intepretation, harmonic functions and their properties, fundamental solution. Poisson's equation, Green's functions and the representation formula for the solutions to the corresponding classical Dirichlet problems.
Sobolev spaces: definitions, weak derivatives, properties as Banach spaces, embedding theorems, boundary values of Sobolev functions (trace theorem), Poincaré inequality.
Elliptic equations: elliptic operator in divergence and in non-divergence form, equivalence for operators with smooth coefficients, weak formulation of boundary value problems for operators in divergence form, existence of solutions, introduction to elliptic regularity, operators in non-divergence form: Hopf lemma and the maximum principle.
Heat equation: statement of the problem, physical derivation, fundamental solution, initial value problems, Duhamel's principle, heat balls and maximum principle in bounded domains, uniqueness for the mixed problem, nonuniqueness without growth conditions for the initial value problem, infinite propagation speed, energy methods, smoothness of classical solutions.
Wave equation: statement of the problem, physical derivation, d'Alembert's formula for the 1-d initial value problem for the homogeneous wave equation, mixed problem for the 1-d homogeneous wave equation, Kirchhoff's formula for the 3-d initial value problem for the homogeneous wave equation, Poisson's formula for the 2-d initial value problem for the homogeneous wave equation, remarks on the higher dimensional case, Huygen's principle, Duhamel's principle and the solution of a generic initial value problem for the wave equation in dimension 1,2,3, energy methods.
|Evans, L. C.
||Partial Differential Equations
||American Mathematical Society
The exam consists in an oral examination but students can access this only after having received a positive evaluation for the resolution of the exercises that will be released at the end of the course.
More precisely, two typologies of exercises will be assigned to the students. The first one consists in "complementary" exercises: students will be asked to complete some computations that will not be done explicitly during the lectures or to prove easy lemmas not requiring the new material introduced in the course but only their background knowledge. These exercises are meant to develop the computational and argumentative skills required to learn profitably the techniques that will be developed in the course. Students do not have to deliver the solutions of these exercises, which however will be asked during the oral examination. They have, instead, to resolve all the exercises contained in a sheet that will be released at the end of the course and they must deliver the solutions at least 10 days before the date of the oral examination. They will access the oral examination only after the positive evaluation of the exercises (in case of negative evaluation, they can fix the wrong exercises and try again the next session). During the oral exam, student will be questioned about some of the topics of the course, including the proofs detailed during the lectures and the complementary exercises. When necessary, clarifications about the exercises will be asked.