## Summary

# FUNCTIONAL ANALYSIS (SPRING 2016)

### LECTURENOTES

Addendum: Baire category theorem

Weeks 1-10. Notes under construction (last update March 6th 2016);

Assessed Homework 1 [due 18/2]: Week1, Problem 5; Week 2, Problem 8; Week 3, Problem 2 and this Sheet (solution is here; Solutions to the problems in the notes will be posted at the end of the course.)

Assessed Homework 2 [due 17/3] Week 4, Problem 6; Week 6, Problem 3, Week 7: Problem 4 and this: Worksheet on Weak Convergence (file now includes solution)

Solutions to exercises (G. Holzegel and A. Shao)

Revision Sheet (last updated 22/3/16)

# PDE (Spring 2013+2014)

### 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+9.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

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

## WORKSHOP ON HYPERBOLIC PDE -- SEPT 30TH--OCT 2ND 2015

## Publications

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

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