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University of Parma
Course catalogue
Università degli Studi di Parma
course catalogue

ADVANCED ANALYSIS 1
cod. 19052

Academic year 2008/09
1° year of course - First semester
Professor
Academic discipline
Analisi matematica (MAT/05)
Field
Formazione analitica
Type of training activity
Characterising
60 hours
of face-to-face activities
6 credits
hub:
course unit
in - - -
lectures timetable unavailable
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Learning objectives

Aim of the course is to give students an overview on the Sobolev spaces as well as some applications to the study of partial differential equations of elliptic type.

Prerequisites

The course ``Spazi di funzioni''

Course unit content

An overview on L^p spaces.<br />
<div align="justify">Definitions and basic properties. Completeness of L^p. Convolutions. Young's theorem. Approximation of a L^p function with smooth functions. A compactness criterion in L^p.<br />
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Sobolev spaces.<br />
A first motivation to the study of Sobolev spaces. Weak derivatives and comparison with the concept of distributional derivatives. First properties of Sobolev spaces. Friedrichs' theorem. Equivalent characterizations of W^{1,p}(A) spaces. Chain rule. Approximation with smooth functions in R^N. Approximation with smooth functions in an open set A. Lifting operators. Convolutions in W^{1,p}(A). The Sobolev space W^{m,p}(A) (m>1 and integer). The definition of support of a function in L^1_{loc}(A). Definition and main properties of the Sobolev spaces W_0^{m,p}(A). Sobolev inequality and embedding theorems. Poincaré inequality. Definition of the trace of a function in a Sobolev space.<br />
</div>
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Some boundary value problems.<br />
The concept of weak solution, comparison with classical solutions. Variational form of some boundary value elliptic problems. Lax and Milgram theorem. Smoothness of the weak solution.

Full programme

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Bibliography

R.A. Adams, Sobolev spaces, ACADEMIC PRESS, New York, S. Francisco, London, 1975;<br />
H. Brezis, Analisi funzionale, LIGUORI;<br />
D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd Edition, SPRINGER-VERLAG, New York, 1983.

Teaching methods

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Assessment methods and criteria

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Other information

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