## Summary

**PDE (Spring 2020)**

** ****Updated April 5th Revision Sheet for the exam.**

**Coursework 1 (deadline 18/2/2020)**

**Week 1: Problems 4 and 7Week 2: Problems 1 (including showing existence, i.e. the analogue of Exercise 1.1) and 4Week 3: Problems 5 and 8 Solutions_CW1.pdf**

Coursework 2 (deadline to be confirmed)

Week 5: Problems 5 and 6**Week 6/7: Problems 3, 4 and 6SolutionsCW2**

LECTURE NOTESLECTURE NOTES

### WEEK 1.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 and 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

Laplace's equation, Fundamental Solution, Mean value formulas, Poisson's formula, Maximum Principles, Dirichlet problem on a ball

### Week6.PDF

Elliptic Regularity Theory, General 2nd order elliptic equations, Existence of Weak Solutions, Boundary Regularity, Fredholm Alternative; Week 6 and 7 are in the same pdf.

### Week8.PDF

Basic Fourier-Analysis, Schroedinger equation, Non-linear Schroedinger, Heat equation (fundamental solution, boundary initial value problem, maximum principle)

### Week10.PDF

The Wave equation: Energy Estimate, Domain of Dependence, Fourier Synthesis, Kirchhoff's formula, Duhamel formula, existence and uniqueness for perturbations of the wave operator (Week 9 and 10 are in the same pdf.)

**Supplementary Material:**

Distributions (some background material)

Blow-up for semi-linear wave equations (including the proof of a special case of Fritz John's blow-up results)

**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.

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

**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, Luk J, Smulevici J, et al. , 2019, Asymptotic properties of linear field equations in anti-de sitter space, *Communications in Mathematical Physics*, Vol:374, ISSN:0010-3616, Pages:1125-1178

Holzegel G, Dafermos M, Rodnianski I, 2019, Boundedness and decay for the Teukolsky equation on Kerr spacetimes I: the case |a|≪M, *Annals of Pde*, Vol:5, ISSN:2524-5317, Pages:1-118

Dafermos M, Holzegel G, Rodnianski I, 2019, The linear stability of the Schwarzschild solution to gravitational perturbations, *Acta Mathematica*, Vol:222, ISSN:1871-2509

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:1432-0916, Pages:723-775