Politecnico di Torino
Politecnico di Torino
Politecnico di Torino
Academic Year 2017/18
Functional analysis/Partial differential equations
1st degree and Bachelor-level of the Bologna process in Mathematics For Engineering - Torino
Teacher Status SSD Les Ex Lab Tut Years teaching
Berchio Elvise   O2 MAT/05 40 20 0 0 5
Vallarino Maria   O2 MAT/05 40 20 0 0 5
SSD CFU Activities Area context
B - Caratterizzanti
B - Caratterizzanti
Formazione teorica
Formazione teorica
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.
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. Riesz-Frechet theorem. Weak and weak* convergence.
Adjoint operator of a bounded operator on Hilbert spaces. Unitary, normal and self-adjoint operators.
Spectrum of a bounded operator on Hilbert spaces: definition and properties.
Compact operators: definition and examples. Spectrum of a compact self-adjoint operator on Hilbert spaces.

PART II: Partial Differential Equations
Basic concepts: linear, semi-linear, quasi-linear and non linear equations. Classical and weak solutions. Well-posed 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 half-space. Dual spaces.
Weak formulation of elliptic problems: existence, uniqueness, continuous dependence from data.
Spectral theory for self-adjoint 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 one-year 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
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.

Programma definitivo per l'A.A.2017/18

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