Luca ESPOSITO | CALCULATION OF VARIATIONS

cod. 0522200027

CALCULATION OF VARIATIONS

	0522200027
	DEPARTMENT OF MATHEMATICS
	EQF7
	MATHEMATICS
	2024/2025

CURRICULUM APPLICATION-ORIENTED MATHEMATICS

	YEAR OF COURSE 2
	YEAR OF DIDACTIC SYSTEM 2018
	AUTUMN SEMESTER

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/05	6	48	LESSONS	SUPPLEMENTARY COMPULSORY SUBJECTS

EXAMS

Exam	Date	Session
CALCOLO DELLE VARIAZIONI	21/01/2025 - 09:00	SESSIONE ORDINARIA
CALCOLO DELLE VARIAZIONI	21/01/2025 - 09:00	SESSIONE DI RECUPERO
CALCOLO DELLE VARIAZIONI	11/02/2025 - 09:00	SESSIONE ORDINARIA
CALCOLO DELLE VARIAZIONI	11/02/2025 - 09:00	SESSIONE DI RECUPERO
CALCOLO DELLE VARIAZIONI	26/02/2025 - 09:00	SESSIONE ORDINARIA
CALCOLO DELLE VARIAZIONI	26/02/2025 - 09:00	SESSIONE DI RECUPERO

	Objectives
	GENERAL OBJECTIVE: THE TEACHING AIMS TO INTRODUCE STUDENTS TO THE MODERN THEORY OF THE CALCULUS OF VARIATIONS, HIGHLIGHTING THE VERSATILITY OF ITS METHODS IN THE FIELDS OF MATHEMATICS AND PHYSICS (EXISTENCE OF GEODESICS, MINIMAL SURFACES, ISO-PERIMETRIC PROBLEM, EXISTENCE OF SOLUTIONS FOR NON-LINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS). KNOWLEDGE AND UNDERSTANDING: STUDENTS WILL ACQUIRE A THOROUGH UNDERSTANDING OF THE FUNDAMENTAL CONCEPTS OF THE CALCULUS OF VARIATIONS. IN PARTICULAR, THE STUDENT WILL BECOME CAPABLE OF SOLVING SIMPLE PROBLEMS IN VARIOUS AREAS SUCH AS LAGRANGIAN MECHANICS, MATERIALS SCIENCE, POLITICAL ECONOMY, OR MODERN PHYSICS. THE STUDENT WILL BE ABLE TO FORMULATE SIMPLE VARIANTS OF THE THEORETICAL RESULTS LEARNED IN ORDER TO USE THEM IN THE ABOVE-MENTIONED APPLIED CONTEXTS. IN PARTICULAR, THE ABILITY TO REDUCE THE MINIMIZATION OF A FUNCTIONAL TO THE ASSOCIATED EULER EQUATION WILL BE REQUIRED. ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING: STUDENTS WILL BE ABLE TO FORMULATE SIMPLE VARIANTS OF THE THEORETICAL RESULTS LEARNED IN ORDER TO USE THEM IN THE ABOVE-MENTIONED APPLIED CONTEXTS. IN PARTICULAR, THE ABILITY TO REDUCE THE MINIMIZATION OF A FUNCTIONAL TO THE ASSOCIATED EULER EQUATION WILL BE REQUIRED. JUDGMENT AUTONOMY: OBJECTIVE: TO FOSTER THE DEVELOPMENT OF STUDENTS' JUDGMENT AUTONOMY SO THAT THEY CAN CRITICALLY ANALYZE THE CONCEPTS AND THEORIES OF THE CALCULUS OF VARIATIONS. METHODS: TO STIMULATE STUDENTS TO CRITICALLY EVALUATE THE EVIDENCE AND ARGUMENTS PRESENTED DURING THE COURSE AND TO ENHANCE THEIR DECISION-MAKING CAPACITY IN MATHEMATICAL CONTEXTS. COMMUNICATIVE SKILLS: OBJECTIVE: TO IMPROVE STUDENTS' COMMUNICATIVE SKILLS SO THAT THEY CAN CLEARLY EXPRESS THEIR OWN IDEAS AND ARGUMENTS, BOTH ORALLY AND IN WRITING, USING APPROPRIATE TECHNICAL LANGUAGE. METHODS: TO OFFER STUDENTS THE OPPORTUNITY TO PARTICIPATE ACTIVELY DURING CLASSES, ENCOURAGING THEM TO EXPRESS THEIR IDEAS, ASK QUESTIONS, AND PARTICIPATE IN DISCUSSIONS. FURTHERMORE, TO PROMOTE THE WRITING OF PROOFS THAT REQUIRE CLEAR AND WELL-STRUCTURED COMMUNICATION OF MATHEMATICAL IDEAS. LEARNING ABILITY: OBJECTIVE: TO FOSTER STUDENTS' AUTONOMOUS LEARNING SO THAT THEY CAN DEVELOP EFFECTIVE STUDY STRATEGIES, DEEPEN THEIR UNDERSTANDING OF CONCEPTS, AND APPLY THE ACQUIRED KNOWLEDGE CREATIVELY. METHODS: TO PROVIDE RESOURCES AND TEACHING MATERIALS THAT ALLOW STUDENTS TO DEEPEN THEIR COMPETENCE IN CONCEPTS BEYOND CLASSROOM LECTURES, SUCH AS RECOMMENDED READINGS, ADDITIONAL EXERCISES, AND OPTIONAL ADVANCED TOPICS. FURTHERMORE, TO ENCOURAGE ACTIVE PARTICIPATION OF STUDENTS THROUGH CRITICAL ANALYSIS OF MATHEMATICAL PROBLEMS AND INDEPENDENT RESEARCH OF SOLUTIONS.

GENERAL OBJECTIVE:
THE TEACHING AIMS TO INTRODUCE STUDENTS TO THE MODERN THEORY OF THE CALCULUS OF VARIATIONS, HIGHLIGHTING THE VERSATILITY OF ITS METHODS IN THE FIELDS OF MATHEMATICS AND PHYSICS (EXISTENCE OF GEODESICS, MINIMAL SURFACES, ISO-PERIMETRIC PROBLEM, EXISTENCE OF SOLUTIONS FOR NON-LINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS).

KNOWLEDGE AND UNDERSTANDING:
STUDENTS WILL ACQUIRE A THOROUGH UNDERSTANDING OF THE FUNDAMENTAL CONCEPTS OF THE CALCULUS OF VARIATIONS. IN PARTICULAR, THE STUDENT WILL BECOME CAPABLE OF SOLVING SIMPLE PROBLEMS IN VARIOUS AREAS SUCH AS LAGRANGIAN MECHANICS, MATERIALS SCIENCE, POLITICAL ECONOMY, OR MODERN PHYSICS. THE STUDENT WILL BE ABLE TO FORMULATE SIMPLE VARIANTS OF THE THEORETICAL RESULTS LEARNED IN ORDER TO USE THEM IN THE ABOVE-MENTIONED APPLIED CONTEXTS. IN PARTICULAR, THE ABILITY TO REDUCE THE MINIMIZATION OF A FUNCTIONAL TO THE ASSOCIATED EULER EQUATION WILL BE REQUIRED.

ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING:
STUDENTS WILL BE ABLE TO FORMULATE SIMPLE VARIANTS OF THE THEORETICAL RESULTS LEARNED IN ORDER TO USE THEM IN THE ABOVE-MENTIONED APPLIED CONTEXTS. IN PARTICULAR, THE ABILITY TO REDUCE THE MINIMIZATION OF A FUNCTIONAL TO THE ASSOCIATED EULER EQUATION WILL BE REQUIRED.

JUDGMENT AUTONOMY:
OBJECTIVE: TO FOSTER THE DEVELOPMENT OF STUDENTS' JUDGMENT AUTONOMY SO THAT THEY CAN CRITICALLY ANALYZE THE CONCEPTS AND THEORIES OF THE CALCULUS OF VARIATIONS. METHODS: TO STIMULATE STUDENTS TO CRITICALLY EVALUATE THE EVIDENCE AND ARGUMENTS PRESENTED DURING THE COURSE AND TO ENHANCE THEIR DECISION-MAKING CAPACITY IN MATHEMATICAL CONTEXTS.

COMMUNICATIVE SKILLS:
OBJECTIVE: TO IMPROVE STUDENTS' COMMUNICATIVE SKILLS SO THAT THEY CAN CLEARLY EXPRESS THEIR OWN IDEAS AND ARGUMENTS, BOTH ORALLY AND IN WRITING, USING APPROPRIATE TECHNICAL LANGUAGE. METHODS: TO OFFER STUDENTS THE OPPORTUNITY TO PARTICIPATE ACTIVELY DURING CLASSES, ENCOURAGING THEM TO EXPRESS THEIR IDEAS, ASK QUESTIONS, AND PARTICIPATE IN DISCUSSIONS. FURTHERMORE, TO PROMOTE THE WRITING OF PROOFS THAT REQUIRE CLEAR AND WELL-STRUCTURED COMMUNICATION OF MATHEMATICAL IDEAS.

LEARNING ABILITY:
OBJECTIVE: TO FOSTER STUDENTS' AUTONOMOUS LEARNING SO THAT THEY CAN DEVELOP EFFECTIVE STUDY STRATEGIES, DEEPEN THEIR UNDERSTANDING OF CONCEPTS, AND APPLY THE ACQUIRED KNOWLEDGE CREATIVELY. METHODS: TO PROVIDE RESOURCES AND TEACHING MATERIALS THAT ALLOW STUDENTS TO DEEPEN THEIR COMPETENCE IN CONCEPTS BEYOND CLASSROOM LECTURES, SUCH AS RECOMMENDED READINGS, ADDITIONAL EXERCISES, AND OPTIONAL ADVANCED TOPICS. FURTHERMORE, TO ENCOURAGE ACTIVE PARTICIPATION OF STUDENTS THROUGH CRITICAL ANALYSIS OF MATHEMATICAL PROBLEMS AND INDEPENDENT RESEARCH OF SOLUTIONS.

	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).

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 48 HOURS OF FRONTAL LESSON DEVIDE IN 40 HOURS OF THEORETICAL LESSONS AND 8 HOURS OF EXERCISE. 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.

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

Lessons Timetable

BETA VERSION Data source ESSE3 [Ultima Sincronizzazione: 2024-11-18]

Luca ESPOSITO | CALCULATION OF VARIATIONS

CALCULATION OF VARIATIONS

CURRICULUM APPLICATION-ORIENTED MATHEMATICS

TEACHING UNITS

PROFESSORS

EXAMS

SHEET

-

Luca ESPOSITO | CALCULATION OF VARIATIONS