# Analysis III Autumn 2018

Lecturer
Alessio Figalli
Coordinator
Xavier Fernández-Real

## Exam Revision

The exam revision will take place on 28.2 and 4.3 from 8-9 AM, in the room HG G19.1.

## Exam Information

The exam will take place on 23.01, at 14.00-17.00 PM, in HIL G41.

Students will be able to take the exam either in German or in English.

Electronic devices (such as, but not limited to, smartphones, calculators, tablets, laptops, etc.) are not allowed to use during the exam. Students are allowed to bring course notes to the exam, either printed or hand-written. Students are also allowed to bring summaries of the course notes, or summarised formula sheets. Students are NOT allowed to bring the exercises and the solutions from this course to the exam (either printed, or hand-written). Students can also bring formula sheets, such as standard solutions to ODE, typical basis in which to represent solutions to PDEs, fourier series, etc.

## Time and Room

The lecture will take place at 8-10a.m. on Monday in HG E5. The first lecture will be on 24.09.

For the exercise classes, see the section Exercise classes.

## 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

Date Summaries of the lectures Chapters of the Literature Comments
24.9 Classification of PDEs, examples, associated conditions to obtain a unique solution. 1.1, 1.2, 1.3, 1.6, 1.5.1, 1.5.2, 1.4.3. -
1.10 First order equations, quasilinear equations, the method of characteristics, examples. 2.1, 2.2, 2.3 (up to Example 2.2). -
8.10 Examples of the characteristics method, and the existence and uniqueness theorem. 2.3, 2.4 (examples 2.3, 2.4, 2.5, 2.6), 2.5 (existence and uniqueness theorem). -
15.10 Conservation laws and shock waves. 2.7 (up to example 2.14, included). -
22.10 Shock waves: the Rankine-Hugoniot condition, and the entropy condition. Classification of second order linear PDEs. 2.7, 3.2. -
29.10 The one-dimensional wave equation, canonical form and general solution. The Cauchy problem and d'Alembert formula. 4.2, 4.3. -
5.11 Domain of dependence. The non-homogeneous one-dimensional wave equation. Nonhomogeneous d'Alembert formula. Separation of variables. 4.4, 4.5, 5.2 (up to (5.14)). -
12.11 Separation of variables for the heat and wave equation. Dirichlet and Neumann boundary conditions. 5.2 (up to (5.26)), 5.3 (except Example 5.2). -
19.11 Separation of variables for non-homogeneous equations. Resonance. The energy method for the wave and heat equation, and uniqueness of solutions. 5.4, Example 6.45, 5.5 (Examples 5.3 and 5.5). -
26.11 Elliptic equations. The weak maximum principle. The mean value principle. The strong maximum principle. 7.1, 7.2, 7.3 (except Theorem 7.9). -
3.12 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 (Theorem 7.14), 7.6, 7.7 (7.7.1). -
10.12 Separation of variables in rectangles, Dirichlet and Neumann compatibility conditions. The Laplace equation in circular domains. 7.7.1 (Examples 7.22, 7.24, 7.25), 7.7.2 (Laplace eq. in the unit disk). -
17.12 The Laplace equation in circular domains: annulus and sectors. A taste of the Calculus of Variations. Overview of the course. 7.7.2 (7.26, 7.27, 7.28). -

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, or during the corresponding Exercise class.

The students are also encouraged to attend the Präsenz offered by D-MATH assistant group 6. The last regular Präsenz during the semester is on 19.12.

Exercise sheet Due by Solutions
Exercise sheet 1 Fri 28.9 Solutions 1
Exercise sheet 2 Fri 5.10 Solutions 2
Exercise sheet 3 Fri 12.10 Solutions 3
Exercise sheet 4 Fri 19.10 Solutions 4
Exercise sheet 5 Fri 26.10 Solutions 5
Exercise sheet 6 Fri 2.11 Solutions 6
Exercise sheet 7 Fri 9.11 Solutions 7
Exercise sheet 8 Fri 16.11 Solutions 8
Exercise sheet 9 Fri 23.11 Solutions 9
Exercise sheet 10 Fri 30.11 Solutions 10 (corrected on 6.12)
Exercise sheet 11 Fri 7.12 Solutions 11
Exercise sheet 12 Fri 14.12 Solutions 12
Exercise sheet 13 Fri 21.12 Solutions 13
Extra exercise sheet (winter break) - Solutions extra exercise sheet.

The students can choose the preferred exercise class to attend. The first exercise class will be on 28.9.

ZeitRaumTutorSprache
Fr 10-12CAB G 56Jacob Claryssede
Fr 10-12CLA E 4Philipp Fischerde
Fr 10-12ETZ E 7Leon Rigonide
Fr 10-12ETZ G 91Vincent Wüstde
Fr 10-12ETZ H 91Alvin Pyngottyde
Fr 10-12HG F 26.3Gabriel Voirolen
Fr 10-12LEE C 114Frederic Odermatten
Fr 10-12LFW B 3Manuel Fritschede