Luca ESPOSITO | CALCULATION OF VARIATIONS
Luca ESPOSITO CALCULATION OF VARIATIONS
cod. 0522200027
CALCULATION OF VARIATIONS
| 0522200027 | |
| DEPARTMENT OF MATHEMATICS | |
| EQF7 | |
| MATHEMATICS | |
| 2023/2024 |
| YEAR OF COURSE 2 | |
| YEAR OF DIDACTIC SYSTEM 2018 | |
| AUTUMN SEMESTER |
| SSD | CFU | HOURS | ACTIVITY | |
|---|---|---|---|---|
| MAT/05 | 6 | 48 | LESSONS |
| Objectives | |
|---|---|
| 1. KNOWLEDGE AND UNDERSTANDING: THE COURSE AIMS TO INTRODUCE STUDENTS TO THE MODERN THEORY OF THE CALCULUS OF VARIATIONS BY HIGHLIGHTING THE VERSATILITY OF ITS METHODS IN THE FIELD OF MATHEMATICAL AND PHYSICAL SCIENCES (GEODETICS IN RIEMANNIAN GEOMETRY, MINIMAL SURFACES, THE ISOPERIMETRIC PROBLEM, EXISTENCE OF SOLUTIONS FOR NON-LINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS). 2. ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING: THE STUDENT SHOULD BE ABLE TO FORMULATE SIMPLE VARIANTS OF THE THEORETICAL RESULTS LEARNED IN ORDER TO USE THEM IN THE APPLICATION CONTEXTS INDICATED ABOVE. IN PARTICULAR, THE ABILITY TO REFER THE MINIMIZATION OF A FUNCTIONAL TO THE ASSOCIATED EULER EQUATION WILL BE REQUIRED. |
| Prerequisites | |
|---|---|
| THE COURSE ASSUMES KNOWLEDGE OF THE EXAM CONTENTS OF ISTITUZIONI DI ANALISI SUPERIORE |
| Contents | |
|---|---|
| 1) INTRODUCTION: NEWTON'S EQUATION OF MOTION LAGRANGIAN AND HAMILTONIAN FORMALISM. BRACHISTOCHRONE PROBLEM. GEODETIC. ELECTROSTATICS. MINIMUM AREA SURFACES. (4 HOURS) 2) DIRECT METHODS AND THE EXISTENCE OF MINIMUM: WEAK GRADIENT AND SOBOLEV SPACES (2 HOURS). REGULARIZATION OF THE FUNCTIONS OF SOBOLEV AND CONSEQUENCES (2 HOURS). LOWER SEMICONTINUITY AND CONVEXITY (4 HOURS). DIRECT METHODS IN THE CLASS OF LIPSCHITZIAN FUNCTIONS. MEYERS-SERRIN THEOREM (4 HOURS). MORREY AND SOBOLEV THEOREMS. DIRECT METHOD IN SOBOLEV SPACES (4 HOURS). EULER-LAGRANGE EQUATION (2 HOURS). THEOREMS OF EXTENSION, APPROXIMATION AND COMPACTNESS ON REGULAR OPEN OPEN (2 HOURS). POINCARÉ INEQUALITIES. VALUES ON BOARD AND TRACE OPERATOR (2 HOURS). 3) REGULARITY OF MINIMA: MINIMIZATION IN SOBOLEV AND MINIMIZATION IN C^1 (4 HOURS). MINIMUM REGULARITY. ELLIPTIC EQUATIONS FOR THE DERIVATIVES OF MINIMA (4 HOURS). ELLIPTIC EQUATIONS WITH HOLDERIAN COEFFICIENTS (4 HOURS). ELLIPTIC EQUATIONS WITH MEASURABLE COEFFICIENTS (4 HOURS). INTERNAL REGULARITY FOR MINIMA OF UNIFORMLY CONVEX FUNCTIONALS (6 HOURS). |
| Teaching Methods | |
|---|---|
| THE COURSE INCLUDES FRONTAL THEORETICAL LESSONS, DURING WHICH THE WAY IN WHICH THE ACQUIRED KNOWLEDGE CAN BE USED FOR THE SOLUTION OF PROBLEMS CONNECTED TO THE TOPICS COVERED WILL ALSO BE ILLUSTRATED. PARTICIPATION IN THE FRONTAL TEACHING IS STRONGLY RECOMMENDED. |
| Verification of learning | |
|---|---|
| THE EXAM CONSISTS OF AN ORAL TEST AIMED AT ASSESSING THE WHOLE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED IN THE LESSONS WITH CONCEPTUAL AND TECHNICAL QUESTIONS ON THE TOPICS CONSIDERED IN THE LESSONS. DURING THE ORAL EXAMINATION THE CANDIDATE WILL ALSO BE ASKED TO CARRY OUT AN EXERCISE OF THE SAME TYPE AS THOSE CARRIED OUT IN LESSON. HONOR MAY BE GIVEN TO STUDENTS WHO DEMONSTRATE THEY ARE ABLE TO INDEPENDENTLY APPLY THE KNOWLEDGE AND SKILLS ACQUIRED EVEN IN CONTEXTS DIFFERENT FROM THOSE PROPOSED IN LESSON. |
| Texts | |
|---|---|
| -ENRICO GIUSTI, METODI DIRETTI NEL CALCOLO DELLE VARIAZIONI. UNIONE MATAMATICA ITALIANA -ANTONIO AMBROSETTI, APPUNTI SULLE EQUAZIONI DIFFERENZIALI ORDINARIE. SPRINGER -FILIP RINDLER, CALCULUS OF VARIATIONS. SPRINGER -LUIGI AMBROSIO, ALESSANDRO CARLOTTO, ANNALISA MASSACCESI, LECTURE ON ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS. SPRINGER |
| More Information | |
|---|---|
| WEB: HTTPS://DOCENTI.UNISA.IT/003512/RISORSE |
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