
Course unit
FUNCTIONS THEORY
SCP3050963, 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 
8.0 
Course unit organization
Period 
First 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 
Prerequisites:

Besides the courses of Analysis 1 and 2, the courses of Real Analysis and Functional Analysis 1 
Target skills and knowledge:

Familiarity with the main spaces of functions and generalized functions. Mastery of the standard techniques in the theory of distribution, Sobolev spaces and functions with bounded variation. Familiarity with the language of Geometric Measure Theory. Ability of using the knowledge provided by the course for the solution of simple problems in Mathematical Analysis. 
Examination methods:

Home exercises (one exercise sheet for each of the four parts of the course), according to which a mark will be proposed to the student. An oral examination is optional. 
Assessment criteria:

Mastery of the acquired knowledge and ability in utilizing it for the solution of simple problems. Completeness and clarity of the solutions to the proposed exercises (also of theoretical type). In case of oral examination, mastery of the proofs exposed in the course. 
Course unit contents:

Between brackets we denote topics that might be skipped or exposed without proofs according to time availability and/or audience interests.
THEORY OF DISTRIBUTIONS
Definitions, derivatives in the sense of distributions, order of a distribution, compactly supported distributions, convolutions, tempered distributions, Fourier transform, applications.
SOBOLEV SPACES
Definition and elementary properties, approximation theorems, boundary trace and extension results, SobolevGagliardoNirenberg, PoincarĂ© and Morrey inequalities, compactness theorems, [capacity and fine properties of Sobolev functions].
ELEMENTS OF GEOMETRIC MEASURE THEORY
Recap of some measure theoretical tools, covering theorems and differentiation of measures, Hausdorff measure and dimension, Lipschitz functions and Rademacher theorem, rectifiable sets, approximate tangent space, [area and coarea formulae].
FUNCTIONS WITH BOUNDED VARIATION
Definition, approximation and compactness results, [trace and extension theorems], coarea formula, sets with finite perimeter, [isoperimetric inequalities, reduced boundary and structure theorem for sets with finite perimeter, fine properties and decommposability of the derivative of a BV function] 
Planned learning activities and teaching methods:

Blackboard lessons. 
Additional notes about suggested reading:

Possible references not included in the reference texts will be directly recommended in the classroom. 
Textbooks (and optional supplementary readings) 

Bony, JeanMichel, AnalyseJ.M. Bony. [Paris]: Ecole polytechnique, 1988.

Evans, Lawrence C.; Gariepy, Ronald F., Measure theory and fine properties of functionsLawrence C. Evans and Ronald F. Gariepy. Boca Raton [etc.]: CRC, .

Ambrosio, Luigi, Corso introduttivo alla teoria geometrica della misura e alle superfici minimeLuigi Ambrosio. Pisa: Scuola normale superiore, 1997.

Innovative teaching methods: Teaching and learning strategies
 Loading of files and pages (web pages, Moodle, ...)
Innovative teaching methods: Software or applications used
 Moodle (files, quizzes, workshops, ...)

