34686 - Fundamentals of Advanced Analysis 2

Learning outcomes

The course aims at giving the student the fundamentals of the theory of distributions, Sobolev spaces and abstract measure theory. At the end of the course the student will be able to independently study abstract and application

Course contents

The course is organized in two parts. Part 1 (prof. F. Ferrari) Variational inequalities, Variational inequalities in Hilbert spaces  with particular attention to Sobolev spaces and the obstacle problem. Part 2 (prof. A. Parmeggiani) Schwarz class of rapidly decreasing functions. Temperate distributions. Fourier transform of Schwartz functions and temperate distributions. Applications to partial differential equations.

Readings/Bibliography

Part 1. Some of the topics will be discussed following the following books: D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications Academic Press 1980, R. Adams, Sobolev Spaces Academic Press 1975. In addition might be useful also: L.C. Evans, R.F. Gariepy, Measure theory and ne properties of functions, Studies inAdvanced Mathematics, CRC Press, Boca Raton, Flo. (1992). W. Ziemer, Weakly dierentiable functions: Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics 120, Springer-Verlag, New York (1989) Part 2. 1) L. Grafakos: Classical Fourier Analysis, Graduate Texts in Mathematics 249, 2nd Edition, Springer 2) G. Eskin: Lectures on Linear Partial Differential Equations, Graduate Studies in Mathematics Vol. 123, American Mathematical Society Further reading: 3) G. Grubb: Distributions and Operators, Graduate Texts in Mathematics 252, Springer 4) J. Rauch: Partial Differential Equations, Graduate Texts in Mathematics 128, Springer-Verlag 5) G. Folland: Introduction to Partial Differential Equations. Second Edition. Princeton University Press.

Teaching methods

Lectures at the blackboard and advanced seminars of the students.

Assessment methods

The student can choose between a traditional oral exam and a 45 minutes seminar on an advanced topic related to the topics of the course that has not been developed within the course itself. Some exercises will be proposed to the students during the course and the discussion of their solutions will be part of the final exam.

Office hours

See the website of Fausto Ferrari

See the website of Alberto Parmeggiani