(in Polish) Przestrzenie Sobolewa między rozmaitościami Riemanna

elective monographs

The course assumes previous knowledge from the courses of Introduction to partial differential equations (including some familiarity with weak derivatives) and Mathematical analysis II.1 and II.2. Basic knowledge of differential geometry and functional analysis will also be useful.

Sobolev spaces between Riemannian manifolds are not linear spaces. From the topological point of view these spaces are much richer than the classical Sobolev spaces, in particular they have a rich structure of homotopy classes. A basic tool of PDEs is the density of smooth functions in Sobolev spaces. This fact fails in the case of mappings into manifolds.

During this course, we will explore the basic issues, similarities and differences with classical Sobolev spaces: approximation theory and homotopy theory within Sobolev spaces, as well as the trace theory and the lifting theory in the framework of Sobolev spaces.

Various physical or biological phenomena can be modeled by functions which are solutions to partial differential equations. In nature we observe singularities, like in a vortex formed when one drains water from a sink, where the singularity occurs at the center of the spin corresponding to the high velocities at that point. Thus, in order to set a model properly we must consider solutions that are not necessarily continuous but merely in a Sobolev space. The choice of the space is a part of the model. For many problems which appear naturally in different areas, like physics or geometry, the framework of Sobolev spaces needs to be restricted to maps whose range is constrained in a manifold.

Among many examples, the simplest illustration seems to be geodesics: given two points on a surface, we look for a path connecting two points with minimal length.

Sobolev spaces between Riemannian manifolds are not linear spaces. From the topological point of view these spaces are much richer than the classical Sobolev spaces, in particular they have a rich structure of homotopy classes. A basic tool of PDEs is the density of smooth functions in Sobolev spaces. This fact fails in the case of mappings into manifolds.

The speed and level of the course will be adjusted to the possibilities of the audience. The exercise classes will partly have a form of a seminar, discussing the geometric and analytic tools used in this area.

The following topics will be covered:

-Definition and basic properties of classical Sobolev spaces.

- Definition and motivation of Sobolev spaces between manifolds. Connections with harmonic maps and other models.

- Approximation theorems, counterexamples.

- Homotopy theory in the framework of Sobolev spaces.

-Gagliardo's trace theorem for classical Sobolev spaces, counterexamples and generalization in the framweork of Sobolev spaces between manifolds.

- Lifting problem for Sobolev mappings.

- Connections between the lifting and the trace problem.

Original papers:

1. Schoen, Richard; Uhlenbeck, Karen A regularity theory for harmonic maps. J. Differential Geometry 17 (1982), no. 2, 307–335.

2. Schoen, Richard; Uhlenbeck, Karen Boundary regularity and the Dirichlet problem for harmonic maps. J. Differential Geom. 18 (1983), no. 2, 253–268.

3. Hang, Fengbo; Lin, Fanghua Topology of Sobolev mappings. II. Acta Math. 191 (2003), no. 1, 55–107.

4. Bourgain, Jean; Brezis, Haïm; Mironescu, Petru Lifting, degree, and distributional Jacobian revisited. Comm. Pure Appl. Math. 58 (2005), no. 4, 529–551.

Oral exam. In order to be able to take the oral exam one must give at talk durint the exercise classes.