Instructor. Dr. Marta Lewicka (office hours in Thack
408, Monday, Wednesday 11:00-12:00)
Prerequisites. The course is intended for graduate students;
undergraduates require permission of the instructor (easily granted).
Familiarity with Real Analysis at the graduate level will be assumed.
If in doubt, consult with the books:
Brezis: Functional analysis, Sobolev spaces and Partial
Differential Equations (also, you may read the shorter French
edition: Analyse fonctionelle - Theorie et applications)
Lieb and Loss: Analysis
Grades. Grades will be based on two midterms (25% +
25%) and a presentation (50%).
Incompletes and make-up exams will almost never be given,
and only for cases of extreme personal tragedy. The same applies to
change to 'audit' after the fourth week of instruction:
it will be permitted only for a very
good reason ("I just realized that this course is too difficult for
me" is not a good reason).
Topics.
The purpose of this course is to introduce a variety of
modern techniques and results in Partial Differential
Equations, Calculus of Variations and Analysis.
The course will be self-contained.
List of topics:
1. Existence of solutions to the nonhomogeneous steady Navier-Stokes
equations (after C. Amick)
2. Young measures and compensated compactness (after L. Tartar and F. Murat)
3. A compactness result in the gradient theory of phase transitions
(after A. Desimone, S. Muller, R. Kohn and F. Otto).
4. Korn's inequality (a detailed discussion including the
proof after Kondratiev and O. Oleinik) and the nonlinear rigidity
estimate (after G. Friesecke, R. James and S. Muller).
5. Existence, regularity and wellposedness of the Cauchy
problem in the inviscid and the viscous hyperbolic systems of
conservation laws in 1 space dimension.
6. Quasiconvexity and partial regularity in the Calculus of
Variations (after L. Evans).
Calendar of presentations:
11 Apr: "Rank-one convexity does not imply quasiconvexity" by
Sverak. Link to paper.
Presentation by Laura Bufford.
16 Apr: The truncation theorem in Appendix 1 of the paper "A
theorem on geometric rigidity and the derivation of nonlinear plate
theory from three dimensional elasticity" by Friesecke, James and
Muller.
Link to paper.
Presentation by Guoqing Liu.
18 Apr: How to remove the unnecessary assumptions in the proof
of partial regularity of minimizers to strongly quasiconvex energies. From
"Quasiconvexity and partial regularity in the Calculus of Variations"
by Evans. Link to paper.
Also see: Link to Mingione's review paper.
Presentation by Pablo Ochoa.
23 Apr: "Line energies for gradient vector fields in the plane" by Ambrosio, De Lellis and Mantegazza. Link to paper.
Presentation by Jilong Hu.
Calendar.
9 Jan (M): First class
16 Jan (M): Martin Luther King Day - no class
5 March, 7 March (M, W): Spring break - no class
11 Oct (Tue): Class from Monday
19 Mar (M) : First Midterm
25 April (W) : Second Midterm and last class