# Instruction on Workshop on Analysis and Geometry Part 1

## Preface

An Instructional Workshop on Analysis and Geometry was held at the Australian National University between the 23rd January and the 10th February, 1995. Over one hundred people attended, with more than twenty universities represented. The majority of participants were Australian PhD students, but there were also students from Germany, New Zealand, Japan and Taiwan. Twenty-five Australian and over-seas speakers participated.

The rationale behind the Workshop was to help rectify an incompleteness in the coursework material available to most mathematics research students in Australia. It was intended that someone attending the introductory first week of the workshop would be exposed to many of the ideas and techniques fundamental to modern analysis and geometry, while those who attended all three weeks would gain an overview up to current research levels. The Workshop was designed with research students in mind, but was also intended for established mathematicians working in other fields.

The workshop consisted of a number of broad streams, each starting at an introductory level and becoming more specialised in the later weeks. There were five one-hour lectures each day.

Notes were prepared by the speakers and were distributed prior to the lectures. The strong demand for these notes has led us to collect them here in three volumes, which we have organised on a subject basis. Volume One deals primarily with Partial Differential Equations, Volume Two with Differential Geometry and Geometric Measure Theory, and Volume Three covers topics from Harmonic Analysis and Nonlinear Analysis.

The Workshop was sponsored by:

- The School of Mathematical Sciences at the ANU, in conjunction with its 1994-5 Special Program on Nonlinear Analysis.
- The Australian Mathematical Soceity through the miniconference fund.
- The Graduate School at the ANU.

We would like to thank the speakers for their efforts, both at the Workshop and in the preparation of these lecture notes.

Tim Cranny and John Hutchinson

(Editors)

## Contents

Chapter | Page no |
---|---|

1 - Measure Theory and Fuction Spaces (PDF, 2150KB) John Hutchinson |
1 |

2 - Second Order Linear Partial Differential Equations (PDF, 2662KB) John Urbas |
39 |

3 - Finite Element Method for Numerically Solving PDE's (PDF, 931KB) Steve Roberts |
77 |

4 - The Dirichlet Problem for the Minimal Surface equation (PDF, 1638KB) Graham Williams |
91 |

5 - Elliptical Systems (PDF, 771KB) John Hutchinson |
111 |

6 - Nonclassical Solutions of Fully Nonlinear Elliptic PDE's (PDF, 1536KB) Tim Cranny |
121 |

7 - Inverse Spectral Problems (PDF, 2150KB) Richard Melrose |
137 |

## Printed copy

## Volume

## Date

## Venue

## Copyright

First published in Australia 1996

© Centre for Mathematics and its Applications

Mathematical Sciences Institute

The Australian National University

CANBERRA ACT 0200

AUSTRALIA

This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.

Edited by

Tim Cranny

John Hutchinson

Instructional workshop on Analysis and Geometry

ISBN 0 7315 2414 4 (set)

ISBN 0 7315 2415 2 (part 1)

ISBN 0 7315 2416 0 (part 2)

ISBN 0 7315 2417 9 (part 3)