MA4013: Advanced topics in Analysis with Application to PDEs

School Cardiff School of Mathematics
Department Code MATHS
Module Code MA4013
External Subject Code G100
Number of Credits 20
Level L7
Language of Delivery English
Module Leader Professor Federica Dragoni
Semester Spring Semester
Academic Year 2015/6

Outline Description of Module

Partial differential equations (PDEs) are used to model almost everything around us. This module provides a broad overview on modern analysis techniques (e.g. Sobolev spaces, jets, inf-sup convolutions, variational approaches and others) for the theory of linear and nonlinear PDEs.

The aim of this module is not to show how to find an explicit solution for the equation but to introduce advanced analysis tools which will allow us to study a PDE, and many related problems, without looking for any explicit solution.

In particular we will focus on different notions of generalized solutions. In fact, classical solutions are intuitive but many PDEs (in particular nonlinear first-order) do not have any classical solution, thus the need of using more advanced analysis ideas to study them.

For (linear and nonlinear) divergence form equations, we will use the theory of Sobolev spaces to study well-posed problems. For nonlinear first-order and elliptic/parabolic second-order PDEs we will instead consider the modern theory of viscosity solutions.

In the second part of the module will study control problems and differential games and their relations with Hamilton-Jacobi-Bellman equations.

Recommended Modules: MA4007 Measure Theory (in particular the Lebsgue integral will be used to define Sobolev spaces)

On completion of the module a student should be able to

The students should gain some knowledge of each of the following topics:

  • LP and Sobolev spaces
  • Weak (in the distributional sense) solutions for divergence form PDEs
  • Theory of viscosity solutions
  • Control problems and differential games

How the module will be delivered

30 - 50 minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 120 hours of private study, including preparation of worked solutions for problem classes.

Skills that will be practised and developed

  • Use of Sobolev spaces to study divergence form PDEs.
  • Use of viscosity theory.
  • Understanding of the rigorous mathematical formulation of a control problem/differential games and their connection with the theory of nonlinear PDEs.

 

Transferable Skills:

  • Knowledge of several advanced analysis techniques and ideas.
  • Ability to work with PDEs which are used to describe many different problems from many scientific areas.
  • Ability to use control theory and differential games, which have important applications in engineering and finance.

How the module will be assessed

Formative assessment is carried out in the problem classes.  Feedback to students on their solutions and their progress towards learning outcomes is provided during these classes.  

The summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has a choice of four from five equally weighted questions.

Assessment Breakdown

Type % Title Duration(hrs)
Exam - Spring Semester 100 Advanced Topics In Analysis With Application To Pdes 3

Syllabus content

  • The classical theory: harmonic functions, mean value property, maximum principles and comparison principles.
  • Elliptic equations in divergence form and example of non-existence for classical solutions.
  • Sobolev spaces and weak derivatives.
  • Weak solutions for Dirchlet problems with vanishing boundary condition.
  • Non- existence of classical solutions for non-divergence form equations: the eikonal equation.
  • Viscosity solutions: definitions, properties, Perron’s method and comparison principles.
  • Convexity and semiconvexity with applications to inf-sup convolutions.
  • Control system and Hamilton-Jacobi-Bellman equations.
  • Differential games.

Essential Reading and Resource List

See Background Reading List

Background Reading and Resource List

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

Bardi M. and Capuzzo-Dolcetta I., Optimal Control Theory and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhauser


Copyright Cardiff University. Registered charity no. 1136855