Academic Year 2021-2022

HIGHER ANALYSIS

Teachers

Lorenzo Freddi
Guglielmo Feltrin
Unit Credits
6
Teaching Period
Second Period
Course Type
Supplementary
Prerequisites. Obligatory propedeuticity: none.  It is recommended the knowkedge of the subjects of the course of Istituzioni di Analisi Superiore.
Teaching Methods. Lectures
Verification of Learning. Oral exam. Oral presentation of a related subject chosen by the student. 
More Information. On the elearning platform elearning.uniud.it the students can found: – lecture notes.
Objectives
The student will have to:

Knowledge and understanding:

Know the basic concepts presented in the course.

Ability to apply knowledge and understanding:

Know how to apply the theoretical elements in the resolution of specific problems,

such as maximum and / or minimum problems, ordinary or partial differential equations.

Autonomy of judgment:

Know how to locate the most appropriate techniques in solving assigned problems or applications, even outside the specific context of the field of mathematical analysis.

Communicative Skills:

Self-compiling mathematical proofs; introduce, in an oral and written way, a subject, or a mathematical theory, from those learned when attending the course.

Learning ability:

Study independently, starting with the recommended bibliography.

Address the proposed problems by selecting the most meaningful ones independently.

Contents
The course aims to introduce students to one or more of the Mathematical Analysis fields that have developed since the end of the nineteenth century. Modern analysis has a great intrinsic cultural value, lends itself to many applications, and is a natural prerequisite for those who want to continue math studies.

In details, the program of the course may include the following arguments (whenever not yet treated in other courses):

– Functional analysis: regularization and approximation methods by convolution of integrable functions; distribution theory; Sobolev spaces; bounded variation functions.

– Calculus of Variations: the direct method, lower semicontinuity and coercivity for integral functionals; first variation and Euler equations; constrained minimization and Lagrange multipliers; relaxation and variational convergence.

– Optimal control: well posedness and optimality necessary conditions; epidemic optimal control; relaxation and variational convergence of optimal control problems.

– Differential calculus in normed vector spaces: Fréchet derivative, Gateaux derivative, local inversion theorem, global inversion theorem, applications to differential equations.

– Fixed point theorems: Brouwer fixed point theorem, Rothe theorem, Bohl theorem, Poicarè-Miranda theorem, Schauder theorem, applications to differential equations.

– Introduction to topological degree and bifurcation theory, with applications to ecological and celestial mechanics models.

Texts
Bibliographic references will be provided by the teacher based on the program being developed.