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TOPICS IN ADVANCED ANALYSIS 1 (526SM)

A.Y. 2020 / 2021

Professor 
Period 
First semester
Credits 
6
Duration/Length 
48
Type of Learning Activity 
Programme specific subjects
Study Path 
[PDS0-2018 - Ord. 2018] common
Syllabus 
Teaching language 

English

Learning objectives 

KNOWLEDGE AND UNDERSTANDING
By the end of the course the student is expected to be familiar with the most common PDEs and of some of their applications
CAPACITY TO APPLY KNOWLEDGE AND UNDERSTANDING
By the end of the course the student is expected to be able to use the technology presented during the lectures to study the most common PDEs
JUDGMENT AUTONOMY
By the end of the course the student is expected to be able to recognize the basic structure of the most common PDEs and which of tools presented during the course can be used to study their properties (e.g. existence, uniqueness, regularity).
COMMUNICATIVE SKILLS
By the end of the course the student is expected to be able to express himself with proficient command of language and exposure security on the topics of the course.
LEARNING CAPACITY
By the end of the course the student is expected to be able to successfully consult standard texts on PDEs.

Prerequisites 

Differential and integral calculus for functions of several variables, Basic facts on Banach and Hilbert spaces. spaces C^k and L^p

Contents 

1) First order equations:
a. the method of characteristics

b. the Hamilton-Jacobi equation:
i. non existence of smooth solutions
ii. Hopf-Lax formula and comments about the correct notion of solution

c. Conservation laws
i. Non-existence of smooth solutions and non-uniqueness of distributional ones
ii. Lax-Oleinik formula and comments comments about the correct notion of solution

2) Elliptic equations:

a. Laplace equation and harmonic functions:
i. Properties of harmonic functions: mean value property, regularity maximum principle, Harnack inequality
ii. Existence: Perron’s method and the variational approach

b. General elliptic equations:
i. Sobolev spaces
ii. Weak formulation and uniqueness
iii. Existence: Lax-Milgram theorem and Fredholm alternative
iv. Regularity estimates in the interior and at the boundary
v. Maximum principle and Harnack inequality

3) Parabolic equations:

a. Weak formulation and uniqueness
b. Existence: the Galerkin method
c. Maximum principle and Harnack inequality
d. Bits fo semigroup theory and the theorem of Hille-Yosida

Teaching format 

Oral lectures
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Any changes to the methods described here, which become necessary to ensure the application of the safety protocols related to the COVID19 emergency, will be communicated on the websites of the Department of Mathematics and Geoscience - DMG and of the Study Program in Mathematics. **********************************************************************

Extended Programme 

.

End-of-course test 

Oral examination.

Other information 

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Texts/Books 

Evans “Partial differential Equations”
Ambrosio, Carlotto, Massaccesi “Lectures on Elliptic Partial Differential Equations”


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