96735 - Advanced Methods of Mathematical Analysis

Learning outcomes

The course aims at giving the student the fundamentals of the theory of Sobolev spaces. At the end of the course the student will be able to independently study abstract and applied theories that require the knowledge of the aforementioned theories.

Course contents

 

The course is organized in two parts.

Part 1 (prof. F. Ferrari): Variational inequalities. Variational inequalities in Hilbert spaces, with particular focus on Sobolev spaces and introductions to free boundary problems. Introduction to the theory of viscous solutions to nonlinear PDEs.

Part 2 (prof A. Parmeggiani): Sovability of partial differential operators acting on Hilbert spaces. Sobolev spaces on Rn, duality between Sobolev space, the Sobolev Embedding Theorem, applications to regularity theory of PDEs. Sobolev spaces on a bounded open set and applications to the spectral theory of the Laplacian with homogeneous boundary data (homogeneous Dirichlet data). Hypoelliptic differential operators. Wave-front set of a distribution.

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 fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, Flo. (1992). W. Ziemer, Weakly differentiable functions: Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics 120, Springer-Verlag, New York (1989)

 

Part 2.

G. B. Folland: Introduction to Partial Differential Equations. Second Edition. Princeton University Press.

F.G. Friedlander. Introduction to the theory of distributions. Second edition. With additional material by M. Joshi. Cambridge University Press, Cambridge, 1998.

L. H\"ormander: The Analysis of Linear Partial Differential Operators, I. Reprint of the second (1990) edition. Classics in Mathematics. [https://mathscinet.ams.org/mathscinet/search/series.html?id=3749] Springer-Verlag, Berlin, 2003

Teaching methods

Lectures at the blackboard and advanced seminars of the students.

Assessment methods

Students may 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.

Teaching tools

See the website of Fausto Ferrari

 

See the website of Alberto Parmeggiani

Office hours

See the website of Fausto Ferrari

See the website of Alberto Parmeggiani