PDE (Spring 2020)
Updated April 5th Revision Sheet for the exam.
Coursework 1 (deadline 18/2/2020)
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
Coursework 2 (deadline to be confirmed)
Week 5: Problems 5 and 6
Week 6/7: Problems 3, 4 and 6
PDE examples, ODE review, Picard's theorem, Gronwall's inequality, bootstrap technique
First order PDEs, method of characteristics, Cauchy Problem, Burger's equation, weak solutions (idea and examples)
Cauchy Kovalevskaya Theorem, Holmgren's uniqueness theorem
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.
Laplace's equation, Fundamental Solution, Mean value formulas, Poisson's formula, Maximum Principles, Dirichlet problem on a ball
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.
Basic Fourier-Analysis, Schroedinger equation, Non-linear Schroedinger, Heat equation (fundamental solution, boundary initial value problem, maximum principle)
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.)
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).)
ESI SummER SCHOOL Vienna 2014
Quick Intro to Lorentzian Geometry (Edinburgh 2017)
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