Online: 2021, September 7 (2h), 8 (2h), 10 (4h), 14 (4h), 16 (4h), 17 (2h), 20 (2h), 21 (1h), 22 (1h), 23 (2h), 24 (2h), 27 (2h), 28 (2h), 30 (2h)
This course will cover the basic theory of Sobolev Spaces in the Euclidean Space and in Riemannian manifolds. Some more advanced topics will be also introduced, when necessary for the applications, which will refer to problems in unbounded domains of the Euclideanspace orin non-compact Riemannian manifolds.The proposed duration of the course is 2 months, with 16 hours of lectures per month.
1) Sobolev spaces. Basic theory. Approximation by smooth functions.Results of compactness. Embeddings. Poincaré and Hardy inequali-ties. Isoperimetric and Faber-Krahn inequalities. Interplay betweenthe geometry of the manifold and the embedding results.
2) Linear and non-linear diffusion equations; the concept ofsolutionsand the variety of possible behaviors. The energy method. Regulariza-tion and iterative techniques.
3) Asymptotics for large times: classical results in the Euclidean space.The asymptotic profile in linear and nonlinear diffusion. The case of the Neumann problem in subdomains of the Euclidean space. Asymptotic behavior in manifolds.