## Summary

**PDE (Spring 2018)**

**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

### 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)

### 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

**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, Shao A, 2017, Unique continuation from infinity in asymptotically anti-de Sitter spacetimes II: Non-static boundaries, *Communications in Partial Differential Equations*, Vol:42, ISSN:0360-5302, Pages:1871-1922

Holzegel G, Shao A, 2016, Unique Continuation from Infinity in Asymptotically Anti-de Sitter Spacetimes, *Communications in Mathematical Physics*, Vol:347, ISSN:0010-3616, Pages:723-775

Holzegel G, 2016, Conservation laws and flux bounds for gravitational perturbations of the Schwarzschild metric, *Classical and Quantum Gravity*, Vol:33, ISSN:0264-9381

Holzegel G, Klainerman S, Speck J, et al. , 2016, Small-data shock formation in solutions to 3D quasilinear wave equations: An overview, *Journal of Hyperbolic Differential Equations*, Vol:13, ISSN:0219-8916, Pages:1-105