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Risorsa bibliografica obbligatoria |
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Risorsa bibliografica facoltativa |
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Anno Accademico
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2015/2016
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Scuola
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Scuola di Ingegneria Industriale e dell'Informazione |
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Insegnamento
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097673 - CALCULUS OF VARIATIONS
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| Docente |
Tomarelli Franco
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| Cfu |
8.00
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Tipo insegnamento
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Monodisciplinare
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| Corso di Studi |
Codice Piano di Studio preventivamente approvato |
Da (compreso) |
A (escluso) |
Insegnamento |
| Ing Ind - Inf (Mag.)(ord. 270) - BV (478) NUCLEAR ENGINEERING - INGEGNERIA NUCLEARE | * | A | ZZZZ | 097673 - CALCULUS OF VARIATIONS | | Ing Ind - Inf (Mag.)(ord. 270) - MI (487) MATHEMATICAL ENGINEERING - INGEGNERIA MATEMATICA | * | A | ZZZZ | 097673 - CALCULUS OF VARIATIONS |
| Programma dettagliato e risultati di apprendimento attesi |
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Aims and scope
The course provides an introduction to modern techniques in Calculus of Variations and Geometric Measure Theory. Variational formulations of boundary value problems for differential equations and problems with unilateral constraints are studied in connection with minimization of integral functionals. Some applications to continuum mechanics and image analysis are presented.
PROGRAM
1. Dido's problema. Steiner's problem. First variation of an integral functional.
2. Classical Calculus of Variations. Euler-Lagrange equations. First integrals. Null-lagrangian integrals. The direct method in Calculus of Variations.
3. Isoperimetric inequality. Wirtinger inequality. Laplace equation. Mean value property. Bounded slope condition. Hilbert Theorem concerning Lipschitz extremals.
4. Friedrichs mollifiers. Sobolev spaces. Poincaré inequality. Traces. Convexity of integral functionals. Semicontinuity of integral functionals. Sobolev embeddings.
5. Reflexivity. Separability. Weak coinvergence. Weak-star coinvergence. Banach-Alaoglu-Bourbaki Theorem. Elliptic equations in divergence form. Clamped plate. Elasticity system.
6. Variational inequalities in Hilbert spaces. Lions-Stampacchia Theorem. Minimum problems with convex constraints. Noncoercive problems. Recession cone. Obstacle problem for membrane and plate.
7. Measures. BV functions of one or several variables. Absolutely continuous functions.
8. Hausdorff measure. Hausdorff dimension. Hutchinson's self-similar fractal sets. Rectifiable sets. Federer-Vol'pert Theorem. Cantor-Vitali function.
9. Finite perimeter sets. First variation of area functional and mean curvature. Mumford-Shah functional: strong and weak formulation.
10. Gamma-convergence. Definitions and examples.
PREREQUISITI
Basic courses on Mathematical Analysis of Laurea Triennale in Ingegneria.
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| Note Sulla Modalità di valutazione |
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The final exam consists in an oral exposition about the whole program, with closer examination on two chapters to be chosen in the program. The choice must be communicated before the exam.
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Luigi Ambrosio, Nicola Fusco, Diego Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Editore: Clarendon Press, Anno edizione: 2000, ISBN: 9780198502456
Heidy Attouch, Giuseppe Buttazzo, Gérard Michaille, Variational Analysis in Sobolev and BV Spaces, Editore: MOS-SIAM, Anno edizione: 2014, ISBN: 978-1-611973-47-1
Note:Second Edition
Haim Brezis, Analisi funzionale, Editore: Liguori, Anno edizione: 1986, ISBN: 978-88-207-1501-4
Stefan Hildebrandt, Anthony Tromba, The Parsimonious Universe, Editore: Springer, Anno edizione: 1996, ISBN: 978-0387979915
Mariano Giaquinta, La forma delle cose, Editore: Edizioni di Storia e Letteratura, Anno edizione: 2010, ISBN: 9788863721782
David Kinderlehrer, Guido Stampacchia, An Introduction to Variational Inequalities and Their Applications, Editore: SIAM, Anno edizione: 2000, ISBN: 0-89871-466-4
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| Nessun software richiesto |
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Tipo Forma Didattica
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Ore didattiche |
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lezione
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50.0
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esercitazione
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20.0
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laboratorio informatico
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0.0
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laboratorio sperimentale
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0.0
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progetto
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0.0
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laboratorio di progetto
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0.0
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| Informazioni in lingua inglese a supporto dell'internazionalizzazione |
Insegnamento erogato in lingua
Inglese
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Disponibilità di libri di testo/bibliografia in lingua inglese
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Possibilità di sostenere l'esame in lingua inglese
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