Partial Differential Equations

Recommended Prerequisites

Functional Analysis.

Teaching Methods

The classes are essentially of expository nature and should include examples and exercises that lead the students to understand and apply 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. Some tutorial support will be available to help the students solving the proposed tasks.

Learning Outcomes

The course is an introduction to the study of partial differential equations (PDEs) using functional analysis and energy methods. Questions of existence, uniqueness and regularity for weak solutions to linear elliptic and parabolic PDEs will be emphasized. Various nonlinear PDEs will also be studied, using a variety of different approaches, like variational and monotonicity methods, fixed-point theorems or intrinsic scaling.

Work Placement(s)

No

Syllabus

A crash course on Sobolev spaces. Second order linear elliptic equations (existence of weak solutions; regularity in the interior and up to the boundary; maximum principles; Harnack inequality; De Giorgi-Nash-Moser theory). Second order linear parabolic equations (existence via Galerkin method; regularity theory and maximum principles). The Calculus of Variations (Euler-Lagrange equation; existence of minimizers; regularity; unilateral constraints: variational inequalities and free boundary problems). Nonvariational techniques (monotonicity and fixed point methods). Degenerate and singular PDEs (the p-Laplace equation; intrinsic scaling; the infinity Laplacian).

Head Lecturer(s)

José Miguel Dordio Martinho de Almeida Urbano

Assessment Methods

Assessment
Synthesis work: 20.0%
Frequency: 30.0%
Exam: 50.0%

Bibliography

H. Brezis, Analyse Fonctionnelle, Masson, 1983.

L.C. Evans, Partial Differential Equations, 2nd edition, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, 2010.

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer, 1983.

Q. Han and F. Lin, Elliptic Partial Differential Equation, Courant Lecture Notes in Mathematics, Vol. 1, American Mathematical Society, 1997.

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, 1980.

P. Lindqvist, Notes on the p-Laplace equation, University of Jyvaskyla, 2005.

J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, 1969.

J.M. Urbano, The Method of Intrinsic Scaling, Lecture Notes in Mathematics, vol. 1930, Springer, 2008.