**Room: ** 441 Altgeld

**Time: ** TTh 11:00 am - 12:20 pm

** Lecturer: ** Eduard-Wilhelm Kirr

- Office: 245 Illini Hall
- E-mail: ekirr@math.uiuc.edu
- Phone: 265-5418
- Office hours:
- Tuesdays 4:00 - 5:00 pm
- Wednesdays 12:00 - 1:00 pm
- By appointment

**Syllabus:**

- 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.
- 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.
- Semigroups of operators. 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.

**Prerequisites: ** Real analysis (Math 447 or equivalent) and multivariable calculus. Familiarity with theory of linear operators and with partial differential equations (Math
553 or 554) would be helpful.

** References: **

- L. Nirenberg:
*Topics in Nonlinear Functional Analysis* - L.C. Evans:
*Partial Differential Equations* - A. Pazy:
*Semigroups of linear operators and applications to partial differential equations* - 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:

- Notes from Lecture 1 in mht format and in pdf format.
- Notes from Lecture 2 in mht format and in pdf format and Inequalities Handout.
- Notes from Lecture 3 in mht format and in pdf format .
- Notes from Lecture 4 in mht format and in pdf format .
- Notes from Lecture 5 in mht format and in pdf format .
- Notes from Lecture 6 in mht format and in pdf format .
- Notes from Lecture 7 in mht format and in pdf format.
- Notes related to variational methods:
- Lecture 16 in mht format and in pdf format.
- Lecture 17 in mht format and in pdf format. and handout on convex functions and on applications to nonlinear elliptic propblems
- Lecture 18 in mht format and in pdf format.
- Lecture 19 in mht format and in pdf format.
- Lecture 20 in mht format and in pdf format.

- Notes related to semigroup of operators:
- Notes from Lecture 11 in mht format and in pdf format.
- Notes from Lecture 12 in mht format and in pdf format.
- Notes from Lecture 13 in mht format and in pdf format.
- Notes from Lecture 14 in mht format and in pdf format.
- Notes from Lecture 15 in mht format and in pdf format.