Math 595 BVM Spring 2009

Bifurcation and Variational Methods in Nonlinear Partial Differential Equations

Room: 141 Altgeld

Time: TTh 10:30 - 11:50 am

Lecturer: Eduard-Wilhelm Kirr

• Office: 219 Altgeld Hall
• E-mail: ekirr@math.uiuc.edu
• Phone: 265-5418
• Office hours:
  • Tuesdays 4:00 - 4:00 pm
  • Wednesdays 4:00 - 5:00 pm
  • By appointment

Syllabus:

1. Implicit and inverse function theorem in Banach spaces. Application to existence, uniqueness and bifurcation of solutions of nonlinear elliptic PDE's. Basic theory is in Section 2.7 of L. Nirenberg: ``Topics in Nonlinear Functional Analysis", the application will be taken mostly from recent papers dealing with models in optics, statistical physics and molecular chemistry.
2. Semigroups of operators (review). Applications to existence, uniqueness and stability of solutions of nonlinear evolution PDE's. Basic theory is in L.C. Evans "Partial Differential Equations" or A. Pazy: ``Semigroups of linear operators and applications to partial differential equations". Applications will be mainly from the latter and recent papers.
3. Calculus of Variations concentration compactness and applications to existence, uniqueness and stability of solutions of nonlinear PDE's. Basic theory of calculus of variations is in L.C. Evans "Partial Differential Equations", concentration compactness is in T. Cazenave: ``Semilinear Schroedinger Equations" the applications will come from the second book and papers.

Prerequisites: The course will attempt to be self-contained but familiarity with real analysis (Math 447 or equivalent), multivariable calculus, theory of linear operators and with linear partial differential equations (Math 553 or 554) would be very helpful.

References:

1. L. Nirenberg: Topics in Nonlinear Functional Analysis
2. L.C. Evans: Partial Differential Equations
3. A. Pazy: Semigroups of linear operators and applications to partial differential equations
4. T. Cazenave: Semilinear Schroedinger Equations

Lecture Notes: The mht format has colors and other enhancements like an ActiveX control that you should allow for easy navigation. If your browser cannot open it use the pdf file:

1. Notes from Lecture 1 in mht format and in pdf format.
2. Notes from Lecture 2 in mht format and in pdf format .
3. Notes from Lecture 3 in mht format and in pdf format .
4. Notes from Lecture 5 in mht format and in pdf format . Also slides related to the first project can be found here
5. Notes from Lecture 6 in mht format and in pdf format .
6. Notes from Lecture 7 in mht format and in pdf format.
7. Handout on Convex Functions and Applications to Nonlinear Elliptic Problems on bounded domains.
8. Notes from Lecture 8 in mht format and in pdf format .
9. Notes from Lecture 9 in mht format and in pdf format .
10. Notes from Lecture 10 in mht format and in pdf format.
11. Notes from Lecture 11 in mht format and in pdf format.
12. Notes from Lecture 12 in mht format and in pdf format.
13. Notes from Lecture 13 in mht format and in pdf format.
14. Notes from Lecture 14 in mht format and in pdf format.
15. Notes from Lecture 15 in mht format and in pdf format.