MEASURE AND INTEGRATION (FALL 2016)
The main text for the course is "Real Analysis" by E. Stein & R. Shakarchi (Princeton University Press). The first example sheet is available via
The second sheet is available replacing "measuresheet1.pdf" by "measuresheet2.pdf" in the above link.
FUNCTIONAL ANALYSIS (SPRING 2016)
LECTURENOTES (Notes under construction (last update March 6th 2016).)
PDE (Spring 2013+2014)
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 + 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.
Distributions I, Fundamental Solution, Laplace's equation, Mean value formulas, Poisson's formula, Maximum Principles
[updated 21/3/2014] Dirichlet problem on a ball, Elliptic Regularity Theory, Existence of Weak Solutions
Boundary Regularity, Fredholm Alternative; Week 6 and 7 are in the same pdf.
[updated 29/3/2014] Basic Fourier-Analysis, Schroedinger equation, Non-linear Schroedinger, Heat equation (fundamental solution, boundary initial value problem, maximum principle)
[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.
ESI SummER SCHOOL Vienna 2014
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
Holzegel G, Warnick CM, 2015, The Einstein-Klein-Gordon-AdS system for general boundary conditions, Journal of Hyperbolic Differential Equations, Vol:12, ISSN:1793-6993, Pages:293-342
Holzegel GH, Warnick CM, 2014, Boundedness and growth for the massive wave equation on asymptotically anti-de Sitter black holes, Journal of Functional Analysis, Vol:266, ISSN:0022-1236, Pages:2436-2485
Holzegel G, Smulevici J, 2014, QUASIMODES AND A LOWER BOUND ON THE UNIFORM ENERGY DECAY RATE FOR KERR-ADS SPACETIMES, Analysis & Pde, Vol:7, ISSN:1948-206X, Pages:1057-1090
Holzegel G, Smulevici J, 2013, Decay Properties of Klein-Gordon Fields on Kerr-AdS Spacetimes, Communications On Pure and Applied Mathematics, Vol:66, ISSN:0010-3640, Pages:1751-1802