MA 388: Topics in Non-linear Functional Analysis
Credits: 3:0
In this course, our main aim is to develop abstract variational techniques which can be employed to study the existence of solutions of various Semi-linear elliptic Partial Differential Equations. The main fundamental results, that will be covered in this course, are functional analytic in nature and can be used in many other situations. A basic outline of the course is as follows:
- The Pohozaev identity and non-existence of solutions.
- Calculus in normed linear space: Fréchet and Gâteaux differentiability, Notion of integral, Existence and uniqueness theorem for ODE in Banach space.
- Dirichlet’s principle, Basics of Sobolev spaces, Connection between critical points and solutions of PDE
- Direct Methods in Calculus of Variations:Existence of extrema, Ekeland’s Variational Principle, Constrained critical points (method of Lagrange Multiplier).
- Deformation and the Palais-Smale condition, Saddle points and min-max methods: The mountain pass theorem and its application, The concentration compactness lemmas and their application.
- Additional topics (to be covered if time permits): Linking Theorem,Index Theory, The Brezis-Nirenberg Problem.
Suggested books and references:
- Ambrosetti, A, and Malchiodi, A, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2007.
- Struwe, Michael, Variational methods, Applications to nonlinear partial differential equations and Hamiltonian systems, Second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 34. Springer-Verlag, Berlin 1996.