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ISTITUZIONI DI ANALISI PER LE APPLICAZIONI

Teaching Staff

ALFONSO VILLANI - MODULO 1 - MAT/05 - 6 CFU
GIUSEPPE DI FAZIO - MODULO 2 - MAT/05 - 6 CFU

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Learning Objectives

• MODULO 1
  

  The aim of the course is to make the students familiar with basic concepts, main theorems and most used techniques in Measure and Integration Theory. This will give the students a more complete education in the field of Mathematical Analysis and will provide them with useful prerequisites in order to follow more advanced courses.

• MODULO 2
  

  The primary goal of the course is to get involved in some theoretical problems concerning Partial Differential Equations.

Course Structure

• MODULO 1
  

  The course main topics will be explained by the teacher during formal lectures. These will be focusing on each topic's general principles and new concepts that have not been studied before. Each's topic's additional resources and subchapters will be presented by turning over groups of students. The goal is to have students develop study autonomy and teaching abilities, skills that are essential for students who want to pursue a career in research or teaching.

• MODULO 2
  

  The topics in the course are exposed in a purely theoretical way although they may be applied to Engineering and Physics. Most of the lectures are given in traditional way. Approximately 20% of class hours will be devoted to exercises. Then they may try to solve the exercises even trough collaboration among them. This is because the students may experiment the difficulty of the topics and from collaboration they get the tools to overcome the difficulty.

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Detailed Course Content

• MODULO 1
  

  Lebesgue measure. Measures, outer measure and Carathéodory's theorem. Borel sets of a topological space. Borel measures and distribution functions.Completion of a measure space. Measurable funcions. Sets which are not Lebesgue measurable and Lebesgue measurable sets that are not Borel sets. Signed measures. Integration in a measure space. L^p-spaces. Various types of convergence of sequences of measurable functions. Product measure and Fubini's theorem.

• MODULO 2
  

  Classical and generalized solutions of Partial Differential Equations. Regularity topics for elliptic equations.

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Textbook Information

• MODULO 1
  

  1. A. Villani, Appunti del corso di Istituzioni di Analisi Superiore, lecture notes on line

  2. W. Rudin, Real and Complex Analysis, Third edition, Mc Graw Hill

• MODULO 2
  

  Recommended readings are list here

  http://www.dmi.unict.it/%7Edifazio/mate/PDE/libri-di-testo-equadiff.html

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