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## MAMPDE1-Advanced Partial Differential Equations

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:
Current from: 2018/9

Module Convenor: Dr Nikos Katzourakis

Type of module:

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 on Calculus methods, representation of solutions via integrals, etc, are not required for MAMPDE1.

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.

Additional outcomes:

There will be given a quick introduction to Sobolev spaces and weak topologies.

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

Summative assessment- Examinations:

Summative assessment- Coursework and in-class tests:

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.

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

Reassessment arrangements:
Resubmission of the report.

Additional Costs (specified where applicable):
1) Required text books:
2) Specialist equipment or materials:
3) Specialist clothing, footwear or headgear:
4) Printing and binding:
5) Computers and devices with a particular specification:
6) Travel, accommodation and subsistence:

Last updated: 20 April 2018

THE INFORMATION CONTAINED IN THIS MODULE DESCRIPTION DOES NOT FORM ANY PART OF A STUDENT'S CONTRACT.