Analysis 3 Autumn 2021

Mikaela Iacobelli
Lauro Silini


Informations in ETHZ Course Catalogue.

Written Exam

Random checks of the above rules will be made on the day of the exam.

Time and Room

The lecture takes place in room HG G3 every Monday at 08-10 a.m.

Live Streaming is available for members of ETH. A video recording of the lecture is uploaded every Tuesday under password protection.



In this lecture we treat problems in applied analysis. The focus lies on the solution of quasilinear first order PDEs with the method of characteristics, and on the study of three fundamental types of partial differential equations of second order: the Laplace equation, the heat equation, and the wave equation.


The aim of this class is to provide students with a general overview of first and second order PDEs, and teach them how to solve some of these equations using characteristics and/or separation of variables.

Schematic Syllabus

  1. General introduction to PDEs and their classification (linear, quasilinear, semilinear, nonlinear / elliptic, parabolic, hyperbolic)
  2. Quasilinear first order PDEs
  3. Hyperbolic PDEs
  4. Parabolic PDEs
  5. Elliptic PDEs


Y. Pinchover, J. Rubinstein, "An introduction to Partial Differential Equations", Cambridge University Press (12. Mai 2005).


Analysis I and II, Fourier series (Complex Analysis).

Lecture Summaries

Here is a first version of the complete LECTURE NOTES.

Date Summaries Corresponding chapters in the literature Hand-written notes/Extras
27.09 Introduction, classification of PDEs (order, linearity, quasilinearity, homogeneity), examples, associated conditions to obtain a unique solution. 1.1, 1.2, 1.3, 1.4.3, 1.5.1, 1.5.2. Lecture 01/ Extra 01
04.10 First order equations, quasilinear equations, Method of Characteristics, examples. 2.1, 2.2, 2.3 (up to example 2.2 included) Lecture 02/ Extra 02
11.10 Examples of the characteristics method, and the existence and uniqueness theorem. 2.3, 2.4 (examples 2.3, 2.5, 2.6), 2.5 (Theorem 2.10) Lecture 03/ Extra 03
18.10 Conservation laws and shock waves 2.7 Lecture 04
25.10 Shock waves: the Rankine-Hugoniot condition, and the entropy condition. Classification of second order linear PDEs. 2.7, 3.2 Lecture 05
01.11 The one-dimensional wave equation, canonical form and general solution. The Cauchy problem and d'Alembert formula. 4.1, 4.2, 4.3 Lecture 06
08.11 Domain of dependence. The non-homogeneous one-dimensional wave equation. Nonhomogeneous d'Alembert formula. Separation of variables. 4.4, 4.5, 5.2 Lecture 07
15.11 Separation of variables for the heat and wave equation, homogeneous problems. Dirichlet and Neumann boundary conditions. 5.2, 5.3 Lecture 08
22.11 Separation of variables for non-homogeneous equations. Resonance. The energy method for the wave and heat equation, and uniqueness of solutions. 5.4, 5.5 (Example 6.45, 5.3) Lecture 09
29.11 Elliptic equations. The weak maximum principle. The mean value principle. The strong maximum principle. 7.1, 7.2, 7.3 Lecture 10
6.12 ONLY RECORDED Applications of maximum principle (uniqueness). Green's identities. The maximum principle for the heat equation. Separation of variables for elliptic problems. 7.4, 7.5, 7.6, 7.7. Lecture 11
13.12 Separation of variables in rectangles, Dirichlet and Neumann compatibility conditions. The Laplace equation in circular domains. 7.7.1, 7.7.2 Lecture 12
20.12 🎄 The Laplace equation in circular domains: annulus and sectors. A taste of the Calculus of Variations. Overview of the course. 🎄 7.7.2 Lecture 13


Every Monday after the lecture the corresponding exercise list will be uploaded. We encourage the students to attempt solving the exercises. The first session will be on Friday October 1st . Please do not forget to register to one of the following exercise groups via myStudies.

exercise sheet solutions Comments
Exercise sheet 1 Solutions 1
Exercise sheet 2 Solutions 2 Fixed typo in 2.3: Cross the unique correct answer(s).
Exercise sheet 3 Solutions 3
Exercise sheet 4 Solutions 4
Exercise sheet 5 Solutions 5 Corrected empty reference in part (b), exercise 2. In exercise 5.4 (a) substitute elliptic with parabolic.
Exercise sheet 6 Solutions 6
Exercise sheet 7 Solutions 7 Solution of 7.4 is more detailed now.
Exercise sheet 8 Solutions 8
Exercise sheet 9 Solutions 9
Exercise sheet 10 Solutions 10
Exercise sheet 11 Solutions 11
Exercise sheet 12 Solutions 12
Exercise sheet 13 Solutions 13
MOCK EXAM SOLUTIONS Two typos corrected: ex 1 (c) we ask for the maximal domain where the PDE is elliptic, not just a subset of it. In 1 (f) there was a missing square root in the definition of D.

Exercise classes

Fr 10-12CAB G 56Marina Fernandez Garcia
Fr 10-12CLA E 4Salmane El Messoussi
Fr 10-12ETZ E 7Henry Marius
Fr 10-12ETZ J 91Luca Krebs
Fr 10-12ETZ K 91Anthony Salib
Fr 10-12LEE C 114Jean Mégret
Fr 10-12LFV E 41Eric Soriano Baguet
Fr 10-12onlineJan-Marco Haldemann
Fr 10-12onlineArielle Rüfenacht