Module Provider: Mathematics and Statistics
Number of credits: 16 [8 ECTS credits]
Level:7
Terms in which taught: Spring term module
Pre-requisites:
Non-modular pre-requisites:
Co-requisites: MAMCDTU Data & Uncertainty MAMCDTS Dynamical Systems MAMCDTE Partial Differential Equations MAMCDTN Numerical Methods
Modules excluded:
Module version for: 2016/7

Module Convenor: Dr Nikos Katzourakis

Summary module description:

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, tutorials.

Contact hours:
 Autumn Spring Summer Lectures 20 Tutorials 5 Guided independent study 135 Total hours by term 160.00 Total hours for module 160.00

Summative Assessment Methods:
 Method Percentage Project output other than dissertation 80 Oral assessment and presentation 20

Other information on summative assessment:

Formative assessment methods:
Problem sheets and/or essays

Penalties for late submission:

Where the piece of work is submitted after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

You are strongly advised to ensure that coursework is submitted by the relevant deadine. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Penalties for late submission on this module are in accordance with the University policy. Please refer to page 5 of the Postgraduate Guide to Assessment for further information: http://www.reading.ac.uk/internal/exams/student/exa-guidePG.aspx

Length of examination:

Requirements for a pass:
An average of 50% across the whole module.

Reassessment arrangements:
Resubmission of the report.