# Analysis III Autumn 2017

Lecturer
Alessio Figalli
Coordinator
Pengyu Le

## Time and Room

The lecture will take place at 8-10a.m. on Monday in HG E5. The first lecture will be on 25.09.

For the exercise classes, see the section Exercise classes.

There will be Q&A sessions during the semester, see the section Q&A session.

## Abstract

In this lecture we treat problems in applied analysis. The focus lies on the simplest cases of three fundamental types of partial differential equations of second order: the Laplace equation, the heat equation and the wave equation.

## Prerequisites

Analysis I and II, Fourier series(Komplexe Analysis)

## Literature

• Y. Pinchover, J. Rubinstein, "An introduction to Partial Differential Equations", Cambridge University Press(12. Mai 2005).
• Zusätzliche Literatur:
• Erwin Kreyszig, "Advanced Engineering Mathematics", John Wiley & Sons, Kap. 8, 11, 16 (sehr gutes Buch, als Referenz zu benutzen).
• Norbert Hungerbühler, "Einführung in die partiellen Differentialgleichungen", vdf Hochschulverlag AG an der ETH Zürich.
• G. Felder: Partielle Differenzialgleichungen.

Time Preview of the Lecture Chapters of the Literature Comments
18.12 Course review - -
Time Summaries of the Lecture Chapters of the Literature Comments
25.9 Classification of PDEs, examples, associated conditions to obtain a unique solution 1.1, 1.2, 1.3, 1.4.1, 1.4.2(up to line 2 page 10), 1.5.1, 1.5.2(first half, Dirichlet condition and Newmann condition for the Heat equation) -
02.10 First order equations, quasilinear equations, the method of characteristics, examples 2.1, 2.2, 2.3(up to Example 2.2) -
09.10 Examples of the characteristics method, the existence and uniqueness theorem 2.3, 2.4, 2.5 -
16.10 Conservation laws and shock waves, second-order linear equations in two independent variables: classification, canonical form of hyperbolic equations 2.7(up to example 2.15), 3.1, 3.2, 3.3 -
23.10 The one-dimensional wave equation: canonical form and general solution, the Cauchy problem and d'Alembert's formula, domain of dependence and region of influence 4.1, 4.2, 4.3, 4.4 -
30.10 The Cauchy problem for the nonhomogeneous wave equation, The method of separation of variables: Introduction, Heat equation: homogeneous boundary condition. 4.5(up to example 4.12), 5.1, 5.2(up to page 103 (5.15)) -
06.11 Heat equation: homogeneous boundary condition, separation of variables for the wave equation 5.2, 5.3 -
13.11 Separation of variables for nonhomogeneous equation; Sturm-Liouville problems: Nonhomogeneous equations, Nonhomogeneous boundary conditions 5.4, 6.5 example (6.45), 6.6 example (6.46) Errata: In (6.91) it should be 1/2mπ and not 1/mπ.
20.11 elliptic equations: introduction, basic properties of elliptic problems, the maximal principle, applications of the maximum principle 7.1, 7.2, 7.3, 7.4 -
27.11 elliptic equations: Green's identities, the maximum principle for the heat equation, seperation of variables for elliptic problems 7.5, 7.6, 7.7 (up to page 190) -
04.12 elliptic equations: seperation of variables for elliptic problems 7.7.1 -
11.12 elliptic equations: seperation of variables for elliptic problems, Poisson's formula 7.7.2 (up to Remark 7.29),7.8 -

Students should be prepared and come to the session with concrete questions. The Präsenz offered by D-MATH assistant group 6 will also be helpful.

Time Room Content Assistants
13-14 Tue, 24.10 HG G 19.2 Lectures 1-4 Pengyu Le, Johannes Sager
13-14 Mon, 30.10 HG G 19.2 Lectures 1-5 Leonard Deuschle, Pengyu Le, Manuel Madlener
13-14 Fri, 24.11 HG G 19.2 Lectures 1-8 Timo Laaksonlaita, Pengyu Le

The students could find the exercise box in the room HG F 28. Please hand in the exercise in the corresponding box of your assistant before the deadline showed below.

exercise sheet due by solutions
Exercise sheet 1 Fri 06.10 Solution 1
Exercise sheet 2 Fri 13.10 Solution 2
Exercise sheet 3 Fri 20.10 Solution 3
Exercise sheet 4 Fri 27.10 Solution 4
Exercise sheet 5 Fri 03.11 Solution 5
Exercise sheet 6 Fri 10.11 Solution 6
Exercise sheet 7 Fri 17.11 Solution 7
Exercise sheet 8(New version, updated on 20.11) Fri 24.11 Solution 8
Exercise sheet 9 Fri 01.12 Solution 9
Exercise sheet 10 Fri 08.12 Solution 10
Exercise sheet 11 Fri 15.12 Solution 11
Exercise sheet 12 - Solution 12
Exercise sheet 13 - Solution 13

The students can choose the preferred exercise class to attend.

ZeitRaumTutorSprache
Fr 10-11CLA E 4Leonard Deuschlede
Fr 10-11HG E 33.3Marina Durrerde
Fr 10-11HG F 26.3Michelle Inauende