
Course unit
INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS
INP5070341, A.A. 2019/20
Information concerning the students who enrolled in A.Y. 2019/20
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Core courses 
MAT/05 
Mathematical Analysis 
9.0 
Course unit organization
Period 
First semester 
Year 
1st Year 
Teaching method 
frontal 
Type of hours 
Credits 
Teaching hours 
Hours of Individual study 
Shifts 
Lecture 
9.0 
72 
153.0 
No turn 
Prerequisites:

This course will be completely selfcontained, 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 spacetime. Huyghens principle. Cauchy problem for the heat equation. Properties of the heat semigroup. Laplace equation, sub and superharmonic functions. Koebe's theorem. Hypoellipticity of Laplace equation: the theorem of CaccioppoliCimminoWeyl. Strong maximum principle. Overdetermination and symmetry. The geometry of a beam that undergoes torsion at one of its ends. The soapbubble 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 takehome 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


