MA 374a: Regularity in the Calculus of Variations
Credits: 3:0
Prerequisite courses: MA 221, MA 222, MA 223 (MA 321 will help, but not a prerequisite.)
Dirichlet integral and p-Dirichlet integral: Sobolev spaces, existence of minimizers, weak form of the Euler-Lagrange equations.
- Linear Regularity: L^2 estimate, Schauder estimates, L^p estimates
- DeGiorgi-Nash-Moser theory: Moser iteration, Harnack inequality, DeGiorgi’s solution of Hilbert’s 19th problem.
- Regularity of p-Laplacian: C^{1,\alpha} regularity and Uhlenbeck structure.
Suggested books and references:
- Giaquinta, M., and Martinazzi, L., An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, second ed., vol. 11 of Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)].Edizioni della Normale, Pisa, 2012..
- Giusti, E., Direct methods in the calculus of variations, World Scientific Publishing Co. Inc. River Edge, NJ, 2003..