Соболевские неравенства для «плохих» множеств и весов

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 defined on Riemannian manifolds. Nowadays, this is a vast domain of research with applications to nonlinear partial differential 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.