Calculus of Variations

Vorlesung und Übung, WS 2013

Dozent: Prof. Dr. Caterina Zeppieri

Übung: tba

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Zeit,Ort:	Dienstags, 12.00-14.00 Uhr, M6; Donnerstags, , wöchentlich, Beginn: 09.10.2012, Ende: 31.01.2013
Zuordnung:	Master of Education GG/ BAB/ BK-2F
Anmeldung:	Bitte melden Sie sich für die Vorlesung im HIS an. Die Anmeldung für die Übungen ist ab dem 26.03.2012 im Kursbuchungssystem möglich (tba,tba,tba).
Voraussetzungen:	Measure and Integration Theory Functional Analysis Sobolev Spaces
Hinweis:	Diese Veranstaltung wird auf englisch gehalten.

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Beschreibung:

The central problem of the calculus of variations amounts to finding a solution to (1) where and is a suitable Banach space. The birth of this problem dates back to the seventeenth century. Indeed (1) is connected with classical problems in mechanics, such as the brachistochrone problem (Bernoulli 1697), as well as to classical problems in geometrical optics, such as the ermat principle (Fermat 1662). Despite to its old history, in the last decades there has been a renewed an ever increaseing interest in (1) that was mainly originated by the study of variational models in nonlinear elasticity and, more in general continuum mechanics. The main object of this course will be to study the existence of solutions to (1) in a "reasonable" functional space X. We will focus on two different approaches: the classical and the direct one. More specifically, as in the finite-dimensional case, one way of studying (1) is finding the zeroes of the "derivative" of F, F'(u)=0 known as the Euler-Lagrange equation, and then studying the positivity of the second variation aroud the solutions. to do so there are various necessary or sufficient conditions, namely Weierstrass, Legendre or Jacobi conditions. this is the idea behind the so-called classical method. The direct method deals directly with the functional F; here the main idea is to find conditions on F (and hence on f) that makes it possible to prove the existence of minimizers via a suitable extension of the Weierstrass theorem to the infinite-dimensional setting.

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Aktuelles: Today's (11. Oct) Calculus of Variations lecture didn't take place due to an organizational mistake. Next lecture will take place regularly on Tuesday 16 Oct from 12 to 14 in room M6.

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Literatur:

G. Buttazo, M. Giaquinta, S. Hildebrandt, One-dimensional variational problems. Oxford Lecture Series in Mathematics and its Applications, 15 The Clarendon Press, Oxford University Press, New York, 1998. B. Dacorogna, Introduction to the Calculus of Variations. Translated from the 1992 French original. Second Edition. Imperial ollege Press, London 2009. B. Dacorogna, Direct Methods in the Calculus of Variations. Second Edition. Applied Mathematical Sciences, 78. Springer, New York, 2008.

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Übung:	Übungsblätter:

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Übungsgruppen:	Der Übungsbetrieb startet am tba.10.2012 .
Anmeldung:	Kursbuchungssystem (tba,tba,tba)

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