Math 251A: Introductory PDE

2017 Fall

Material: the topic is mainly elliptic PDEs. The main accent will be on the regularity theory for weak solutions of elliptic equations. We will consider topics as Sobolev regularity, Schauder estimates and Hölder regularity, De Giorgi-Nash-Moser theorem. We will spend some time on fully nonlinear equations and on the Evans-Krylov theorem. If time permits (and the audience is motivated), at the end we will consider transport equations in the context of the Di Perna-Lions-Ambrosio theory.

Instructor: Alpár R. Mészáros, MS 5230 (check the e-mail address on my webpage or on CCLE)

Schedule of classes: MWF - 10-10h50AM

Location: MS 5138

Office hours: by appointment

Main texts: M. Giaquinta and L. Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, 2nd edition, Springer, 2012.

D. Gilbarg and N. Trudinger, Elliptic PDEs of second order, 3rd edition, Springer, 1998.

The textbook of L. C. Evans on PDE is also a useful reference. To review Sobolev spaces and distributions, I recommend for instance the textbook by Evans and Gariepy, and Rudin's Functional analysis, respectively. Other references and notes will be pointed out in due time.

Prerequisites: real analysis and some functional analysis (topics like distributions, Lebesgue, Sobolev and BV spaces, Riesz representation theorem, Lax-Milgram, etc. will be useful, but will be recalled, when needed)

There will be roughly 5 sets of homework sheets assigned (posted on this website). I will require individual work on these assingments with clear and clean arguments, i.e. proper formulations of used theorems, writing detailed proofs, etc. The lack of justification might imply point loss.

Final and grading: there will be no final exam, the final grade will be assigned based on the submitted homework assingments.

Homework sheets