KNOWLEDGE AND UNDERSTANDING
At the end of the course the student will have to demonstrate knowledge of the fundamental object of functional analysis, both linear and nonlinear.
CAPACITY TO APPLY KNOWLEDGE AND UNDERSTANDING
At the end of the course the student must know how to apply the knowledge of basic functional analysis acquired to solve problems and exercises of medium difficulty. The exercises can also be proposed as easy theoretical results.
At the end of the course the student will be able to recognize and apply the most basic techniques of functional analysis and also to recognize the situations and problems in which these techniques can be used advantageously.
At the end of the course the student will be able to express himself appropriately on the topics of the course.
At the end of the course the student should be able to consult the standard texts of functional analysis, both linear and nonlinear.
Basic functional spaces: continuous functions and Lebesgue space; weak topologies and a basic knowledge of the theory of Sobolev spaces.
Spectral theory for compact and self adjoint operators over Hilbert spaces. Applications to Sturm-Liouville type problems. Semigroup theory and Hille-Yosida theorem. Differential calculus in Banach spaces with applications to ODEs. Theory of topological degree, fix point theorems: Brouwer, Schauder. Applications to ODEs (Theorem of Peano). Stability of ODEs.
Lectures and problem sessions. During the course some exercises will be discussed in class.
Linear Analysis. Introduction to functional analysis: Banach spaces (Lp, lp), introduction to the space H1([0,1] and H1(R). Bounded operator, compactness in infinite dimensional spaces, characterization of compactness in spaces of sequences and in Lp spaces. Compact operators. Fredholm’s alternative Theorem. Definition of resolvent and of spectrum of a linear operator, proof that the spectrum is non-empty and closed. Spectrum of self-adjoint operators. Spectral theory for compact and self-adjoint operators. Min-max Fisher-Courant Theorem. Spectral theorem for self-adjoint non-compact operators. Functional calculus. Application to Sturm-Liouville problems: weak formulation, the space H01([0,1], Lax-Milgram Theorem and Stampacchia Theorem, construction of the resolvent map, application to eigenvalues problems, separation of variables method.
Nonlinear Analysis. Differential Calculus in Banach spaces. Gateaux and Frechet differentiable functions, examples. Gateuax differentiability and continuous differential implies Frechet differentiability. Mean value Theorem. Higher order derivatives. Differentiability of Nemiski operators. Inverse function Theorem and Implicit function Theorem. Applications to non-linear problems. Constant rank theorems. Lagrange multiplier Theorem and applications. Bifurcation Theorem, necessary condition of bifurcation, Lyapunov-Schmidt reduction. Bifurcation of the simple eigenvalue. ODE in Banach spaces, differentiable dependence on initial data. Finite dimensional Brower degree: properties, integral formulation and geometric interpretation (linking number). Brower fixed point Theorem, Borsuk Ulam Theorem, constancy of dimension theorem and consequences. Leray Schauder degree, nonlinear compact operators, Schauder fixed point Theorem, Peano Theorem.
The exam program coincides with the arguments of the lectures. The exam will be oral consisting in verifying the comprehension of the contents (definitions and proofs) and the ability in explaining the subject and to correctly apply the theory.
Information about the progress of the program and teaching materials will be posted on the site http://moodle2.units.it
H. Brezis: Functional Analysis. Theory and Applications. Liguori Editore.
A. Bressan: Lecture Notes on Functional Analysis: With Applications to Linear Partial Differential Equations. Graduate Studies in Mathematics Volume: 143; 2013; 250 pp.
A. Ambrosetti, G. Prodi: A primer of nonlinear analysis, Cambridge, Cambridge University Press 1993. VIII, 171 pp.
K. Deimling: Nonlinear Function Analysis, Springer-Verlag Berlin Heidelberg, XIV, 450 pp.