Partial Differential Equations

Year
1
Academic year
2023-2024
Code
02002044
Subject Area
Mathematics
Language of Instruction
Portuguese
Mode of Delivery
Face-to-face
Duration
SEMESTRIAL
ECTS Credits
6.0
Type
Elective
Level
2nd Cycle Studies - Mestrado

Recommended Prerequisites

Topology and Linear Analysis; Real and Functional Analysis.

Teaching Methods

The classes are essentially of expository nature and should include examples and exercises that lead the students to understanding and applying the material being taught. There must be room for the presentation of more elaborated exercises and more detailed constructions of concrete examples. The classes should be focused on the teaching of the reasoning processes, so that the students learn how to manipulate the objects presented along the course and more easily find out by themselves how to reach other results by independent reading or problem solving. Some tutorial support will be available

Learning Outcomes

The course is an introduction to the study of linear partial differential equations (PDEs). The first part is devoted to three classical PDEs: Laplace’s equation, the heat equation and the wave equation. These PDEs, for which explicit solutions are derived, are the prototypes of linear elliptic, parabolic and hyperbolic equations, which will be analyzed in the second part of the course from an abstract perspective. Issues of existence, uniqueness and regularity of weak solutions will be addressed, using variational methods and estimates, introduced as motivation in the first part. The course aims at developing the following skills: knowledge of mathematical results; ability to generalization and abstraction; logic reasoning;   rigorous oral and writing skills; computational ability. On the personal level it also allows to develop self-learning skills and independent thinking.

Work Placement(s)

No

Syllabus

The classical linear PDEs - Laplace’s equation: fundamental solution; Poisson’s equation; mean value formulas; properties of harmonic functions; Green’s function; energy methods. Heat equation: fundamental solution; mean value formula and properties of solutions; energy methods. Wave equation: solution by spherical means; non-homogeneous problem; energy methods. Theory for 2nd order linear PDEs - Sobolev spaces. Elliptic equations: existence of weak solutions; regularity; maximum principles. Parabolic equations: existence of weak solutions; regularity; maximum principles. Hyperbolic equations: existence of weak solutions; regularity; propagation of disturbances.

Head Lecturer(s)

Stéphane Louis Clain

Assessment Methods

Final assessment
Exam: 100.0%

Continuous assessment
Evaluation during the semester includes one or more mid-term exams (75-100%) and presentations, either of proofs of more elaborate results or the solving of problems proposed in the form of homework : 100.0%

Bibliography

L.C. Evans, Partial Differential Equations, American Mathematical Society, 1998.

H. Brézis, Analyse Fonctionnelle, Masson, 1983.

E. DiBenedetto, Partial Differential Equations, Birkhäuser, 1995.

Q. Han, A Basic Course in Partial Differential Equations, American Mathematical Society, 2011.