Monday 14:00 - 16:00 and Tuesday 12:00 - 12:45, ESI lecture hall
starting on October 6, 2003

 
Contents: Sobolev spaces play an outstanding role in modern analysis, in particular, in the theory of partial differential equations and its applications in mathematical physics. They form an indispensable tool in approximation theory, spectral theory, differential geometry etc.. Needless to say the theory of these spaces is of interest in itself being a beautiful domain of analysis.

I include the basics on Sobolev spaces, approximation and extension theorems, imbeddings and compactness theorems, their connection with isoperimetric inequalities, capacities with applications to Sobolev spaces, Besov spaces, boundary traces of Sobolev functions, pointwise inequalities for derivatives, Gagliardo-Nirenberg inequalities, pointwise multipliers, variational approach to elliptic equations, inequalities for general differential operators. Some open problems will be formulated as well.

At the beginning of each lecture, I provide the students with copies of its content. I distribute problems for my students which illustrate my course and we discuss their solutions during 30 minutes each week.

I do not follow any textbook but a part of the material is borrowed from the monographs

• V. Maz'ya "Sobolev Spaces", Springer, 1985
• V. Maz'ya, S. Poborchi "Differentiable Functions on Bad Domains", World Scientific 1997