SM/0187 - PARTIAL DIFFERENTIAL EQUATIONS
Academic Year 2021/2022
Free text for the University
MONICA MARRAS (Tit.)
- Teaching style
- Lingua Insegnamento
|[60/65] MATHEMATICS||[65/40 - Ord. 2020] MATEMATICA PURA||6||48|
With the course the student must acquire the tools and methods for the advanced study of Partial Derivation Equations and their applications. In particular, the student will be able to face problems related to the resolution
and the analysis of Partial Differential Equations and he/she will be able to apply these methods to various problems of scientific interest.
Differential and integral calculus in one and more variables; basic elements
of functional analysis; general topology; Lebesgue measure and integration;
First order linear partial dierential equations. The continuity equation and consequences. The transport equation. Linear, homogeneous transport equation with constant coecients. Linear, non-homogeneous transport equation with variable coefficients. Characteristic curves and explicit solutions.
First order non-linear partial differential equations. Quasilinear equations in special form. Burgers' equation. Comparison principles and uniqueness theorem.
Quasilinear dierential equations of higher order. Power series. Non-characteristic surface. CauchyKovalevskaya theorem.
Second order linear dierential equations. Examples and classification Second order elliptic equations: the Laplace equation. The fundamental
solution. Poisson formula. Green's representation formula. Second order
parabolic equations: the heat equation. The fundamental solution. Green's
representation formula. Second order hyperbolic equations: the wave equation. The homogeneous wave equation on the real line: the d'Alambert solution. The non-homogeneous wave equation on the real line: the Duhamel
principle. The vibrating string equation with fixes ends. The energy method.
The maximum principle and applications to elliptic and parabolicequations. Hopf's principles. Comparison theorems. The optimal maximum principle. The principles of maximum of Nirenberg and Friedman.
Applications of the maximum principle: estimates of the solution and its
gradient in various problems
The course (6 CFU / 48 hours) will consist of lectures, theoretical and applicative, with which the student will obtain the specific cognitive skills of the subject.
Lectures will be prevalently held in classrooms, also integrated with online teaching resources, by using specific online platforms managed by the University of Cagliari
Verification of learning
The exam will aim at evaluating mainly the following items: knowledge of
the course’s contents, independent though, exposition skills. The exam will
consist in two parts:
-write test (15/30).
-An oral exam consisting also in the discussion on write test (16/30).
The final mark will be computed by summing the two marks.
1. L.C. Evans, Partial differential equations, American Mathematical Society, Providence (1998).
2. G. Gilardi, Analisi tre, McGraw-Hill Companies (1994).
Our University provides support to SLD students. Those interested can find more information at the link: http://corsi.unica.it/matematica/info-dsa/.