Scheda programma d'esame
METODI TOPOLOGICI IN ANALISI GLOBALE
|METODI TOPOLOGICI IN ANALISI GLOBALE||MAT/05||LEZIONI||48|
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KnowledgeThe students who successfully complete the course 1)- will be able to demonstrate a solid knowledge of the part of "Degree Theory" in R^N explained during the course (see 1 below), 2)- will be able to demonstrate a solid knowledge of some of the fundamental theorems and techniques of "Nonlinear Analysis" in R^N (see 2 below), 3)- will be able to be master of the subjects they will choose, in accord with the lecturer, among the last part of the matter treated in the course, that is (see 3 below): some further properties of nonlinear maps in R^N, some basic theorems and techniques of "Nonlinear Calculus of Variations" in R^N, some initial and elementary examples (two or three) of nonlinear ordinary differential equations (dynamic elementary systems) that can be approached by previous theorems.
Assessment criteria of knowledgeDuring the oral exam the student is asked to demonstrate an appropriate knowledge of the course material, including the optional part, and his ability in discussing the reading matter thoughtfully and with propriety of expression.
- Final oral exam
The evaluation depends on the final oral exam.
Delivery: face to face
- attending lectures
Syllabus1)- The notion of degree deg(y,A, Omega) will be introduced for continuous maps in R^N defined on the closure of an arbitrary bounded subset Omega of R^N. This procedure will consist in certain number of steps. 2)- The following fundamental theorems, among others, will be studied: - Brouwer fixed point theorem and some extended versions of it, - an arbitrary open bounded subset Omega in R^N is not retractible on its boundary, - if N is even S^N don't admits a C^1 field of nonzero tangent vectors, - Borsuk theorem. 3)- In the last part of the course some optional subjects will be discussed (see 3 in learning outcomes), by means of some theorems proved in the previous part, say: - multiplicative property of the degree in R^N, open map theorem, Jordan theorem in R^N (a fine version by Leray) , - mountain pass theorem, saddle theorem, linking theorem, -Lusternik and Schinerlmann theorem for even functions on S^N, by means of the "genus"- some initial examples (two or three) of nonlinear elementary ordinary differential equations.
BibliographyA file containing lectures and some related topics will be available . Some wider perspectives can be find in the following books: Jacob T. Schwartz, "Nonlinear Functional Analysis", Gordon and Breach, Michael Struwe, "Variational Methods", Springer, Ladyzhenskaya Ural'tseva, "Linear and Quasilinear Elliptic Equations", Academic Press (this last book, edited in the sixties, is very instructive for people who would study nonlinear differential equations by means of topological methods)
Ultimo aggiornamento 14/11/2016 17:27