First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Science
MATHEMATICS
Course unit
FUNCTIONS THEORY
SCP3050963, A.A. 2018/19

Information concerning the students who enrolled in A.Y. 2018/19

Information on the course unit
Degree course Second cycle degree in
MATHEMATICS
SC1172, Degree course structure A.Y. 2011/12, A.Y. 2018/19
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Degree course track GENERALE [010PD]
Number of ECTS credits allocated 8.0
Type of assessment Mark
Course unit English denomination FUNCTIONS THEORY
Website of the academic structure http://matematica.scienze.unipd.it/2018/laurea_magistrale
Department of reference Department of Mathematics
Mandatory attendance No
Language of instruction English
Branch PADOVA
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

Lecturers
Teacher in charge DAVIDE VITTONE MAT/05

ECTS: details
Type Scientific-Disciplinary 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

Calendar
Start of activities 01/10/2018
End of activities 18/01/2019

Examination board
Board From To Members of the board
6 Teoria delle Funzioni - a.a. 2018/2019 01/10/2018 30/09/2019 VITTONE DAVIDE (Presidente)
MARTINAZZI LUCA MASSIMO ANDREA (Membro Effettivo)
CIATTI PAOLO (Supplente)
MARICONDA CARLO (Supplente)
MONTI ROBERTO (Supplente)

Syllabus
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, Sobolev-Gagliardo-Nirenberg, Poincaré 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, Jean-Michel, AnalyseJ.-M. Bony. [Paris]: Ecole polytechnique, 1988. Cerca nel catalogo
  • Evans, Lawrence C.; Gariepy, Ronald F., Measure theory and fine properties of functionsLawrence C. Evans and Ronald F. Gariepy. Boca Raton [etc.]: CRC, --. Cerca nel catalogo
  • Ambrosio, Luigi, Corso introduttivo alla teoria geometrica della misura e alle superfici minimeLuigi Ambrosio. Pisa: Scuola normale superiore, 1997. Cerca nel catalogo

Innovative teaching methods: Teaching and learning strategies
  • Lecturing
  • Loading of files and pages (web pages, Moodle, ...)

Innovative teaching methods: Software or applications used
  • Moodle (files, quizzes, workshops, ...)