## Summary

**PDE (Spring 2018)**

Scroll down for (weekly) lecture notes which also contain problems and exercises.

**Assessment 1** (due February 13th 2018):

Week 1: Problem 3

Week 2: Problems 4 6

Week 3: Problem 1 and Exercise 2.2 in the lecture notes

Solutions

**Assessment 2** (due March 13th 2018):

Sheet2

Solutions

**Revision Sheet with some more problems here.**

**Some hints and solution sketches to problems which have not been assessed can be found in this file. Use at your own risk.**

**LECTURE NOTES**

### WEEK1.PDF

PDE examples, ODE review, Picard's theorem, Gronwall's inequality, bootstrap technique

### Week2.pdf

First order PDEs, method of characteristics, Cauchy Problem, Burger's equation, weak solutions (idea examples)

### WEEK3.PDF

Cauchy Kovalevskaya Theorem, Holmgren's uniqueness theorem

### WEEK4.PDF

Proof of Holmgren's theorem. Fritz John's Global Holmgren theorem, domain of dependence, domain of influence, d'Alembert's formula; Week 3 and 4 are in the same pdf.

Week5.pdf Distributions I, Fundamental Solution, Laplace's equation, Mean value formulas, Poisson's formula, Maximum Principles

### W

Week6.pdf [updated 21/3/2014] Dirichlet problem on a ball, Elliptic Regularity Theory, Existence of Weak Solutions

### WEEK7.PDF

Boundary Regularity, Fredholm Alternative; Week 6 and 7 are in the same pdf.

Week8.pdf [updated 29/3/2014] Basic Fourier-Analysis, Schroedinger equation, Non-linear Schroedinger, Heat equation (fundamental solution, boundary initial value problem, maximum principle)

### W

(updated 21/3/2014) Week9 10.pdf [updated 29/3/2014] The Wave equation: Energy Estimate, Domain of Dependence, Fourier Synthesis, Kirchhoff's formula, Duhamel formula, existence and uniqueness for perturbations of the wave operator

Revision Sheet (updated 21/3/2014)

The Mastery Question will be a question combining different techniques that you have seen. It's a good idea to review the section(s) on the wave equation.

**MEASURE AND INTEGRATION (FALL 2016)**

The main text for the course is "Real Analysis" by E. Stein & R. Shakarchi (Princeton University Press). The example sheets are available here

Sheet1, Sheet2, Sheet3, Sheet4, Sheet5,

Sheet6, Sheet7, Sheet8, Sheet9

There is also a set of Notes which follow very closely the book. They now contain all material that has been lectured. Please email me for typos and corrections.

**Update** (14/1/2017): A small section has been added at the end of the notes which is relevant for **Mastery Students only**.

**Update** (16/3/2017): Some typos in the Example Sheets (7-9) have been corrected.

**Update** (15/5/2017): Correction in Example Sheet 9, Question 2.

Sol1, Sol2, Sol3, Sol4, Sol5, Sol6, Sol7, Sol8, Sol9

The first assessed coursework is due **November 11th**. It consists of Question 1 from Sheet 1, Questions 4 5 from Sheet 2 and Questions 3 4 from Sheet 3. [Returned]

The second assessed coursework is due **December 5th**. It consists of Questions 1 4 from Sheet 5, Questions 1 3 from Sheet 6 and Question 5 from Sheet 7. [Returned]

**FUNCTIONAL ANALYSIS (SPRING 2016)**

LECTURENOTES (Notes under construction (last update March 6th 2016).)

Addendum: Baire category theorem

Revision Sheet (last updated 22/3/16)

## ESI SummER SCHOOL Vienna 2014

Quick Intro to Lorentzian Geometry (Edinburgh 2017)

## WORKSHOP ON HYPERBOLIC PDE -- SEPT 30TH--OCT 2ND 2015

## Publications

### Journals

Holzegel G, Dafermos M, Rodnianski I, A scattering theory construction of dynamical vacuum black holes, *Journal of Differential Geometry*, ISSN:1945-743X

Giulini D, Holzegel G, 2005, Corvino's construction using Brill waves

Holzegel G, 2008, Stability and decay-rates for the five-dimensional Schwarzschild metric under biaxial perturbations

Holzegel G, 2010, Ultimately Schwarzschildean Spacetimes and the Black Hole Stability Problem

Holzegel G, Schmelzer T, Warnick C, Ricci Flow of Biaxial Bianchi IX Metrics, *Classical and Quantum Gravity*, ISSN:1361-6382