Picture of Egor Pifagorov
Миникурс лаб Чебышева: В. Мазья, "Соболевские неравенства для "плохих" множеств и весов"
by Egor Pifagorov - Monday, 12 March 2012, 08:31 PM
Владимир Мазья

05.04, 06.04 -- 17-00, 14-ая линия, ауд. 14
09.04 15-00, ПОМИ, ауд. 311
10.04, 12.04, 13.04 -- 17-00, 14-ая линия, ауд. 14

Соболевские неравенства для "плохих" множеств и весов

Sobolev inequalities for "bad" sets and weights

I survey my old and new results on applications of isoperimetric and
isocapacitary inequalities to the theory of Sobolev spaces.
I began to work on this topic many years ago, when as a fourth
year undergraduate student of mat–mekh I discovered that Sobolev type
inequalities are equivalent to isoperimetric and isocapacitary
inequalities. It turned out that classes of domains and
measures involved in imbedding and compactness theorems could
be completely described in terms of length, area and capacity
minimizing functions. Moreover, without change of proofs, the
same remains true for spaces of functions de?ned on Riemannian
manifolds. Nowadays, this is a vast domain of
research with applications to nonlinear partial di?erential equations,
geometry, spectral theory, Markov processes, and potential theory.
Most results presented in these lectures can be found in my recent book
on Sobolev Spaces with applications to elliptic pdes (Springer, 2011),
where a lot more related information is contained.