# ProfessorGustavHolzegel

Faculty of Natural SciencesDepartment of Mathematics

Professor of Pure Mathematics

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### Location

625Huxley BuildingSouth Kensington Campus

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## PDE (Spring 2020)

Updated April 5th Revision Sheet for the exam.

Week 1: Problems 4 and 7
Week 2: Problems 1 (including showing existence, i.e. the analogue of Exercise 1.1) and 4
Week 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 6
SolutionsCW2

LECTURE 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

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.

## FUNCTIONAL ANALYSIS (SPRING 2016)

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

Revision Sheet (last updated 22/3/16)

## ESI SummER SCHOOL Vienna 2014

Lectures 1-5

Quick Intro to Lorentzian Geometry (Edinburgh 2017)

## 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

More Publications