2. Professors
  3. VITOLO Antonio
  4. Teaching

HIGHER ANALYSIS

Antonio VITOLO HIGHER ANALYSIS

0522200004
DEPARTMENT OF MATHEMATICS
EQF7
MATHEMATICS
2024/2025

YEAR OF COURSE 2
YEAR OF DIDACTIC SYSTEM 2018
AUTUMN SEMESTER
CFUHOURSACTIVITY
648LESSONS
ANTONIO VITOLO T Curriculum
ExamDate
ANALISI SUPERIORE14/01/2025 - 09:00
ANALISI SUPERIORE14/01/2025 - 09:00
ANALISI SUPERIORE28/01/2025 - 09:00
ANALISI SUPERIORE28/01/2025 - 09:00
ANALISI SUPERIORE18/02/2025 - 09:00
ANALISI SUPERIORE18/02/2025 - 09:00
Objectives
THE AIM OF THE COURSE IS TO PROVIDE THE STUDENTS WITH ADVANCED CONCEPTS AND METHODS OF MATHEMATICAL ANALYSIS WHICH ARE ESSENTIAL IN THE DEVELOPMENT AND IN THE APPLICATIONS OF THE MODERN MATHEMATICS.
KNOWLEDGE AND UNDERSTANDING:
THE STUDENTS WILL LEARN THE BASIC ISSUES OF SOBOLEV SPACES, UNDERSTANDING THE MEANING AND MANAGING THE PROOF OF THE MAIN RESULTS.
APPLYING KNOWLEDGE AND UNDERSTANDING: THE STUDENTS WILL BE ABLE TO FORMULATE SIMPLE VARIANTS OF THE LEARNED THEORETICAL RESULTS AND TO GIVE A PROOF, AS WELL AS USE IT TO SOLVE DIFFERENTIAL EQUATIONS AND VARIATIONAL PROBLEMS.
Prerequisites
PREREQUISITES: THEORY OF FUNCTIONS OF ONE AND MORE REAL VARIABLES. LEBESGUE MEASURE AND INTEGRATION IN THE N-DIMENSIONAL REAL SPACE. ELEMENTS OF TOPOLOGY AND METRIC SPACES. BANACH SPACES AND LEBESGUE SPACES. SPAZI DI HILBERT.

PROPAEDEUTIC EXAMS: NONE.
Contents
PROGRAM

1. PRELIMINARIES (2H): ELEMENTS OF TOPOLOGY AND FUNCTIONAL ANALYSIS. SPACES OF CONTINUOUS FUNCTIONS LEBESGUE SPACES.
2. DISTRIBUTIONS (10 H): DEFINITION AND PROPERTIES. DISTRIBUTIONAL DERIVATIVES.
3. SOBOLEV SPACES (20 H): CHARACTERIZATIONS AND PROPERTIES. DENSITY THEOREMS. SOBOLEV INEQUALITIES. SOBOLEV FUNCTIONS WITH ZERO TRACE ON THE BOUNDARY.
4. DIFFERENTIAL EQUATIONS AND VARIATIONAL PROBLEMS (16 H): WEAK SOLUTIONS. REGULARITY. MAXIMUM PRINCIPLE.
Teaching Methods
THE COURSE CONSISTS OF 48 HOURS OF THEORETIC FRONTAL LECTURES (6 ECTS) WITH EXAMPLES AND EXERCISES.
THE ATTENDANCE IS WARMLY RECOMMENDED.
Verification of learning
LEARNING ASSESSMENT WILL BE DONE THROUGH AN ORAL EXAM AT THE END OF THE COURSE, FINALIZED TO EVALUATE THE LEVEL OF STUDENT’S KNOWLEDGE AND UNDERSTANDING, AND THE CAPACITY OF APPLYING IT, BY QUESTIONS ON THE TOPICS COVERED BY THE COURSE. THE STUDENT WILL BE ALSO ASKED TO MAKE A SHORT EXERCISE SIMILAR TO THOSE DONE AT LESSON.

HONORS CAN BE OBTAINED BY A STUDENT WHO EXHIBITS A NOTEWORTHY KNOWLEDGE OF THE ISSUES OF THE COURSE, AS WELL AS HIGH DEGREE OF AUTONOMY IN APPLYING ACQUIRED KNOWLEDGE AND SKILLS ALSO IN NEW CONTEXTS.
Texts
REFERENCE TEXTS:
1. H. BREZIS, ANALISI FUNZIONALE. TEORIA E APPLICAZIONI, LIGUORI, 1986.
2. H. BREZIS, FUNCTIONAL ANALYSIS, SOBOLEV SPACES AND PARTIAL DIFFERENTIAL EQUATIONS, UNIVERSITEXT, SPRINGER, 2011.
3. LECTURE NOTES.

SUGGESTED READINGS:
4. W. RUDIN, FUNCTIONAL ANALYSIS, MC GRAW-HILL, 1991.
5. D. GILBARG AND N.S. TRUDINGER, ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS OF SECOND ORDER, SPRINGER, 2001.
6. L.C. EVANS AND R.F. GARIEPY, MEASURE THEORY AND FINE PROPERTIES OF FUNCTIONS, CRC PRESS, 2015
More Information
email: vitolo@unisa.it
Lessons Timetable

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