Main subject of this course is variational methods (direct method) and their applications to nonlinear PDE.
We explain deformation theory, minimizing method and minimax methods (e.g. mountain pass theorem) and their applications to nonlinear elliptic problems.
In studies of nonlinear problems, analysis in infinite dimensional spaces are very important.
We start with an introduction of fundamental tools in nonlinear analysis in Hilbert spaces (Fréchet derivative etc.) and we explain variational approaches to nonlinear problems, especially we deal with various examples in nonlinear elliptic problems.
- Be familiar with Fréchet derivative and related topics in nonlinear analysis in Hilbert spaces
- Be familiar with variational approaches to nonlinear problems; characterization of solutions of nonlinear elliptic equations
- Understand the deformation theory in Hilbert spaces
- Understand minimizing methods, minimax methods (e.g. mountain pass theorem) in Hilbert spaces
- Applications of minimax methods to nonlinear elliptic problems
variational problems, nonlinear elliptic equations, minimizing methods, minimax methods
|Intercultural skills||Communication skills||Specialist skills||Critical thinking skills||Practical and/or problem-solving skills|
This is a standard lecture courses. Occasionally I will give problems for reports.
|Course schedule||Required learning|
|Class 1||The following topics will be covered in this order: - Prerequisite on functional spaces (basic facts in Sobolev spaces etc.) - Variational characterization of solutions of nonlinear elliptic problems - Basic tools in nonlinear analysis in Hilbert spaces (e.g. Fréchet derivative etc.) - Deformation theory in Hilbert spaces - Minimizing methods, minimax methods (e.g. Mountain pass theorem) - Functionals related to nonlinear elliptic equations and the Palais-Smale condition - Geometry of functionals related to nonlinear elliptic equations and minimax methods - Nonlinear elliptic problems in R^N - Applications to singular perturbation problems (introduction)||Details will be provided during each class session.|
- P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, AMS,1986.
- K. Tanaka, Introduction to variational problems, Iwanami, 2010 (in Japanese).