M4021

M4021 Partial differential equations

Students should discuss with the Honours Coordinator the option of enrolling in PDE topic offered by ICE-EM/AMSI.

Aims: The aim will be to introduce students to standard techniques used in the theory of partial differential equations (PDEs).
Many problems in physics and geometry lead naturally to questions related to PDEs.
In this unit we shall give conditions which guarantee the existence and regularity for solutions of large classes of elliptic and parabolic equations.
The techniques involve function space methods. In particular we shall discuss constructive existence proofs. These form the theoretical basis for numerical schemes such as the finite element method. In this regard the unit may be considered as complementary to applied units on PDEs.

Syllabus: Examples of PDE's:
PDE's from variational integrals; Euler-Lagrange equations

Classical theory of

harmonic functions: subharmonic, superharmonic functions; mean value properties; maximum principles; representation formulae -- Green's functions;
heat equation: heat kernel representation; mean value property; maximum principles.

Functional analytic techniques

Overview of LP-theory; approximation by convolutions.
Weak derivatives; definition of Sobolev spaces; weak solutions.
Embedding theorems; compact embedding.
Review of Hilbert space theory; weak convergence; Riesz, Lax-Milgram theorems.

and, time permitting

Regularity of weak solutions; difference quotient method.
Classical solutions.

Prerequisites: MTH2111 / MTH3111 Mathematical Structures or MTH3132 Analysis and Geometry.

Recommended references:
Adams, R. , Sobolev spaces, Academic Press, 1975.
John, F, Partial differential equations, Springer Verlag, 1978, 3rd edition.
Gilbarg, D., Trudinger, N.S., Elliptic partial differential equations of second order, Springer Verlag, 1983, 2nd edition.

Lecturer: Maria Athanassenas

For current students