Module Provider: Mathematics and Statistics
Number of credits: 10 [5 ECTS credits]
Level:7
Terms in which taught: Spring term module
Pre-requisites: MA3FA1 Functional Analysis I and MA3MTI Measure Theory and Integration
Non-modular pre-requisites:
Co-requisites:
Modules excluded:
Module version for: 2016/7

Module Convenor: Dr Nikos Katzourakis

Summary module description:
This module develops the rudiments of the modern theory of Partial Differential Equations, both linear and nonlinear, which began around the 1930s. The essential tools for PDE theory are Measure Theory and Functional Analysis.

It was realised in the early 20th century that in general it is not possible to write down an explicit formula for the solution. A relevant outstanding problem is that we have to define and study functions without derivates as solutions!

The point of view is that we treat PDEs as being defined by differential operators between Banach spaces and prove existence results by using compactness theorems. Previous study of PDEs which develop the old-fashioned applied approach based of Calculus methods, representation of solutions via integrals, etc, are not required for MA4PDE.

Aims:
This module develops some mathematical theory for linear and nonlinear elliptic equations. Topics to be covered include Sobolev spaces, weak solutions of linear divergence equations, existence/ uniqueness/ regularity theory for 2nd order linear elliptic equations, Calculus of Variations, the Euler-Lagrange equation and system, Null Lagrangians, polyconvexity, lower semicontinuity, existence of minimisers, weak solutions of the Euler-Lagrange equations, Viscosity Solutions of fully nonlinear elliptic equations.

Assessable learning outcomes:
On completion of this module students will have acquired:

•The essential knowledge of the basic modern theory of partial differential equations and of the tools of functional analysis and measure theory involved in the study of PDEs.

Outline content:
Sobolev spaces:

Weak derivatives, functional structure, approximation theorems by smooth functions, trace operators, Gagliardo-Nirenberg-Sobolev estimate, Poincare inequality, Morrey estimate, Rellich compactness theorem, difference quotients.

Linear 2nd order divergence elliptic equations:
Weak solutions, Existence and uniqueness via Lax-Milgram theorem, L^2 regularity theory via a priori estimates.

Calculus of Variations:

the Euler-Lagrange equations, Null Lagrangians, polyconvexity, lower semicontinuity, existence of minimisers, weak solutions of the Euler-Lagrange equations.

Viscosity Solutions of fully nonlinear elliptic equations:

Motivation and main definitions via Jets and via test functions, degenerate ellipticity, the infinity-laplacian of Calculus of Variations in L^infinity.

Brief description of teaching and learning methods:
Lectures supported by problem sheets, tutorials and independent study.

Contact hours:
 Autumn Spring Summer Lectures 20 Tutorials 5 Guided independent study 75 Total hours by term 100.00 Total hours for module 100.00

Summative Assessment Methods:
 Method Percentage Written assignment including essay 80 Oral assessment and presentation 20

Other information on summative assessment:
The essential assessment method of the module is a written report typed with TeX of length minimum 10 and maximum 15 pages which counts for 80% of the total mark. The report will be a small “thesis” with references and abstract and must be relevant to the subject of the lectures.
The topic will be agreed in advance between the lecturer and each individual student, according to the preferences/goal of the student. This could e.g. be an extension of the theory, development of relevant approaches, applications to PDEs, to Harmonic Analysis, to Numerical Analysis, etc.
The assessment method also includes a 10 minute presentation on the subject of the report with 5 minute questions which counts for 20% of the total mark. Successful completion of the presentation is a requirement in order to pass this module.

Formative assessment methods:
The student will be able to submit a draft of their report 2 weeks before the final submission for feedback.

Penalties for late submission:
The Module Convenor will apply the following penalties for work submitted late, in accordance with the University policy.

• where the piece of work is submitted up to one calendar week after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for the piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of five working days;
• where the piece of work is submitted more than five working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

• The University policy statement on penalties for late submission can be found at: http://www.reading.ac.uk/web/FILES/qualitysupport/penaltiesforlatesubmission.pdf
You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Length of examination:
N/A

Requirements for a pass:
A mark of 50% overall.

Reassessment arrangements:
Resubmission of the report in August/September.