
Course unit
CALCULUS OF VARIATIONS
SCP3050978, A.A. 2018/19
Information concerning the students who enrolled in A.Y. 2018/19
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Core courses 
MAT/05 
Mathematical Analysis 
8.0 
Course unit organization
Period 
Second semester 
Year 
1st Year 
Teaching method 
frontal 
Type of hours 
Credits 
Teaching hours 
Hours of Individual study 
Shifts 
Practice 
4.0 
32 
68.0 
No turn 
Lecture 
4.0 
32 
68.0 
No turn 
Examination board
Board 
From 
To 
Members of the board 
6 Calcolo delle Variazioni  a.a. 2018/2019 
01/10/2018 
30/09/2019 
MARTINAZZI
LUCA MASSIMO ANDREA
(Presidente)
VITTONE
DAVIDE
(Membro Effettivo)
MARICONDA
CARLO
(Supplente)
RAMPAZZO
FRANCO
(Supplente)
SORAVIA
PIERPAOLO
(Supplente)

Prerequisites:

The Analysis 1 and 2 and the Real Analysis courses 
Target skills and knowledge:

The classical formalism of the calculus of variations in its hystorical development, with applications and motivations to geometry and physics. The development of the modern theory of the calculus of variations in the setting of Sobolev spaces and the related regulatiry questions. Discussion and solutions of the XIX and XX problems of Hilbert. 
Examination methods:

Homeworks and oral exam 
Assessment criteria:

The teacher will ascertain the student's proficiency in the course's main subjects 
Course unit contents:

Introduction to the classical formalism of the Calculus of Variations: indirect methods, first variation, EulerLagrange equations, applications.
Some examples, including minimal surfaces.
The least action principle and the analytical mechanics of Lagrange.
First direct methods, working in spaces of Lipschitz functions, via a priori gradient estimates.
Modern direct methods: introduction to Sobolev spaces and their use in minimization problems. Tonelli's theorem and the XX problem of Hilbert.
First questions of regularity theory. Regularity of elliptic equations via the Caccioppoli inequality, decay estimates, Campanato spaces.
Some more subtle questions in regularity theory: De Giorgi's solution of the XIX problem of Hilbert. Partial regularity for elliptic systems. 
Planned learning activities and teaching methods:

Blackboard lessons 
Additional notes about suggested reading:

The reference material will be communicated during the course. 
Textbooks (and optional supplementary readings) 

Giaquinta, Mariano; Martinazzi, Luca, <<An >>introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphsMariano Giaquinta and Luca Martinazzi. Pisa: Edizioni della Normale, 2012.


