After successful completion of the course, students are able to formulate and prove regularity results for solutions of elliptic partial differential equations. They have understood the foundations of the proofs.
We discuss selected topic in the regularity theory of elliptic PDEs. Topics include
1) shift theorems in scales of Sobolev spaces
2) regularity theory of elliptic problems in polygons (the techniques of Kondratiev, Grisvard)
3) Morrey and Campanato spaces
4) Holder continuity of solutions of scalar elliptic problems (Theorem of De Giorgi)
The course will be based on (selected chapters) of the books
a) Gilbarg-Trudinger, elliptic partial differential equations of second order
b) Giaquinta-Martinazzi, an introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs
c) Jost, partial differential equations