First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Engineering
Course unit
INP5070341, A.A. 2017/18

Information concerning the students who enrolled in A.Y. 2017/18

Information on the course unit
Degree course Second cycle degree in
IN2191, Degree course structure A.Y. 2017/18, A.Y. 2017/18
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Degree course track Common track
Number of ECTS credits allocated 9.0
Type of assessment Mark
Department of reference Department of Civil, Environmental and Architectural Engineering
E-Learning website
Mandatory attendance No
Language of instruction English
Single Course unit The Course unit can be attended under the option Single Course unit attendance
Optional Course unit The Course unit can be chosen as Optional Course unit

Teacher in charge NICOLA GAROFALO MAT/05

ECTS: details
Type Scientific-Disciplinary Sector Credits allocated
Core courses MAT/05 Mathematical Analysis 9.0

Mode of delivery (when and how)
Period First semester
Year 1st Year
Teaching method frontal

Organisation of didactics
Type of hours Credits Hours of
Hours of
Individual study
Practice 4.0 32 68.0 No turn
Lecture 5.0 40 85.0 No turn

Start of activities 02/10/2017
End of activities 19/01/2018

Examination board
Examination board not defined

Prerequisites: This course will be completely self-contained, and can be profitably followed by any student who has had a good exposure to the fundamentals of calculus of one and several variables. Some of these fundamentals will be recalled in detail during the lectures.
Target skills and knowledge: Fourier transform in the Euclidean space. Solution of the Cauchy problem for the wave equation in the physical space-time. Huyghens principle. Cauchy problem for the heat equation. Properties of the heat semigroup. Laplace equation, sub- and super-harmonic functions. Koebe's theorem. Hypoellipticity of Laplace equation: the theorem of Caccioppoli-Cimmino-Weyl. Strong maximum principle. Overdetermination and symmetry. The geometry of a beam that undergoes torsion at one of its ends. The soap-bubble theorem of A.D. Alexandrov.
Examination methods: The students will be provided with take home written exams of increasing level of difficulty. By taking these exams each student pledges that he/she will work on the test without communicating with any of his/her classmates or
anybody else. Each student is only allowed to discuss the exam with Prof. Garofalo.
Infringement of these rules will be considered academic cheating
and adversely affect the final grade in this course.
Assessment criteria: A final grade will be assigned on the basis of the grades in the take-home exams.
Course unit contents: Partial differential equations (PDEs) are expressions involving an unknown function of two or more variables and a certain number of its partial derivatives. Such equations govern the phenomena of the physical world, and they play a preeminent role both in pure mathematics and in the applied sciences:
1. The small vibrations of the string of a violin are described by the wave equation, a PDE that is ubiquitous in the description of undulatory phenomena.
2. The potential of the gravitational field generated by a certain distribution of mass satisfies (away from the mass itself) a PDE that is known as Laplace equation.
3. The distribution of temperature in a conducting body is described (at least near the source) by yet another PDE known as the heat equation. These are instances of PDEs of linear type.

The principal aim of this course is to bring the audience to mastering some of the basic aspects of PDEs, beginning with the linear models described above. The second part of the course will be devoted to providing the audience with a glimpse into some of the fascinating aspects of nonlinear PDEs.
Additional notes about suggested reading: Lecture notes will be made available to the students.
Textbooks (and optional supplementary readings)
  • Nicola Garofalo, An Introductions to Partial Differential Equations. --: --, 2017. Lecture Notes