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 6Week 6/7: Problems 3, 4 and 6SolutionsCW2LECTURE 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
More Information

Lectures 1-5

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