


Politecnico di Torino  
Academic Year 2017/18  
01RMGMQ Functional analysis/Partial differential equations 

1st degree and Bachelorlevel of the Bologna process in Mathematics For Engineering  Torino 





Subject fundamentals
Functional Analysis can be considered as the extension of linear algebra to infinite dimensional vector spaces. Many integral and differential equations which arise in pure and applied mathematics can be reduced to the study of either linear or nonlinear applications among function spaces, which are infinite dimensional vector spaces. The first part of the course aims to study function spaces and linear applications on such spaces. The second part introduces the students to the study of Partial Differential Equations as an application of the notions of Functional Analysis. The course will focus on boundary and initial value problems for second order linear equations.

Expected learning outcomes
The student will learn the basic concepts of the theory of Banach spaces, Hilbert spaces, Sobolev spaces and bounded linear operators on such spaces. These notions will give the student the mathematical tools to understand and study properly complex mathematical problems. The student will learn to apply the acquired competences in the study of Partial Differential Equations, both in classical and variational sense.

Prerequisites / Assumed knowledge
Mathematical Analysis I and II, Linear algebra and geometry, Complex Analysis, Topology.

Contents
PART I: Functional Analysis
Continuous functions over an interval: completeness of the space C[a,b], Weierstrass approximation theorem. Banach spaces: definition, properties and examples. Spaces l^p and L^p. Riesz Lemma. Compactness of the unit ball in normed spaces. Hilbert spaces: definition and properties, spaces L^2 e l^2. Orthogonal sets and orthonormal basis. Orthogonal projections. Bounded linear operators on normed spaces. Dual and bidual spaces of a normed space. Caracherization of the dual of the spaces l^p and L^p. RieszFrechet theorem. Weak and weak* convergence. Adjoint operator of a bounded operator on Hilbert spaces. Unitary, normal and selfadjoint operators. Spectrum of a bounded operator on Hilbert spaces: definition and properties. Compact operators: definition and examples. Spectrum of a compact selfadjoint operator on Hilbert spaces. PART II: Partial Differential Equations Basic concepts: linear, semilinear, quasilinear and non linear equations. Classical and weak solutions. Wellposed problems. Classification. Laplace equation: harmonic functions, mean property, Poisson formula, fundamental solution and Green function. Distribution and Sobolev Spaces: definition, properties, density theorems, continuous and compact embeddings (Poincaré Inequality, Rellich Theorem, Sobolev embeddings). Prolungation and Trace Theorems in the halfspace. Dual spaces. Weak formulation of elliptic problems: existence, uniqueness, continuous dependence from data. Spectral theory for selfadjoint elliptic operators: abstract formulation and applications. Weak formulation of parabolic and hyperbolic problems: Galerkin approximations, existence, uniqueness, continuous dependence from data. 
Delivery modes
This is a oneyear course organized in two parts. The first semester
will be devoted to Functional Analysis. The second semester will be devoted to Partial Differential Equations. Every part consists in 40 hours of theoretical lessons and 20 hours of exercises lessons. 
Texts, readings, handouts and other learning resources
Texts:
H. Brezis, Analisi funzionale. Teoria e applicazioni. Ed. Liguori. A. Ferrero, F. Gazzola, M. Zanotti, Elementi di Analisi Superiore per la Fisica e l’Ingegneria, Ed. Esculapio. L. C. Evans. Partial Differential Equations, AMS. B. Rynne and M.A. Youngson, Linear Functional Analysis. Notes and exercises will be available on the teaching portal for students attending the course. 
Assessment and grading criteria
The exam aims to test the knowledge of the theoretical notions and the ability to apply them in solving exercises of Functional Analysis and Partial Differential Equations.
The exam consists of a written test that may be completed by an oral exam. WRITTEN TEST: it consists of theoretical questions and exercises about the subjects of the two parts of the program. The written test takes 1 hour and 30 minutes. During the written test it is forbidden to use either books or notes. ORAL EXAM: it concerns the subjects of the program and it might include a discussion of the written test. The oral exam can be requested either by the teacher or by the student (only when the written test is sufficient). The final mark takes into account, in equal measure, of the scores obtained in Functional Analysis and Partial Differential Equations. The maximum mark is 30/30 and the final mark is sufficient if it is greater or equal than 18/30. 
