Laurea magistrale in Matematica

The course aims to provide students with a systematic treatment of the main functional spaces and internal operations that form the indispensable tools for the modern treatment of linear and non-linear differential operators. After infroducing the Fourier transform on the Schwartz class and its dual space of tempered distribution, we will deepen the properties  of the convolution product on various function and distribution spaces. Locally convex topological vector spaces  are also studied, in particular the Fréchet spaces, treating the fundamental example of the Schwartz class. Fourier multipliers are introduced and studied on different function spaces, with application to heat and wave equations.

The teaching provides the basic concepts of the theory of absolutely continuous functions, of the differential calculus in Banach spaces, of the theory of Sobolev spaces and it provides some applications to the study of linear and nonlinear differntial boundary value problems.

The course is suitable for both a program specifically targeted to mathematical analysis, and for programs targeted to subjects where differential operators have relevant applications, such as differential geometry, variety analysis and relativity.

In the 9CFU version, important complements related to geometric and functional inequalities will be performed, with applications to partial differential equations.

The following issues as expected: the knowledge of the classical tools of Fourier analysis, with applications to the Lebesgue and Sobolev aspects and to differential equations; differential calculus in Banach spaces, the knowledge of the class of absolutely continuous functions and BV, of the Schwartz class, of the space of tempered distributions, the fundamental properties of the Sobolev spaces and of  some applications to elliptic boundary value problems.

The exams will take place exclusively in presence, except for exceptions, in accordance with the provisions of the University.

Concernin mosulus A and B, the exam consists of questions related to the theory and proofs expounded throughout the teaching. There may be questions that require the execution of exercises. The exam of modulus C consists in a seminar on some result of the recent literature. The score is expressed as x/30. Foreign students can take the exam in English.

Module A (prof. E. Cordero) 24 hours = 3 cfu

- Locally compact topological vector spaces. Fréchet spaces

- The Schwartz class: main properties, relations with Lebesgue spaces

- The space of tempered distributions: main properties

- Action of the Fourier transform on the Schwartz class and on its dual space of tempered distributions

- Young's convolution inequality, special cases

- Sobolev spaces H^s

-Application of the Fourier trasform to the  Cauchy problem for the heat and wave equations

Module B (prof. A. Boscaggin) 24 hours = 3 cfu

- Differential calculus in Banach spaces: definitions, mean value theorem, local inversion theorem and implicit function theorem

- Lebesgue differentiation theorem and absolutely continuous functions.

- Introductions to Sobolev spaces in one and several variables: definitions, main properties, density of regular functions, extension; embeddings and Poincaré inequality; the homogeneous Dirichlet problem -div (grad u) = f with zero boundary conditions

Module C (prof. S. Terracini) 24 hours = 3 cfu

• Further results and proofs on Sobolev spaces
• Isoperimetric inequality
• BV functions of several variables
• Further results on elliptic problems, maximum principle
• Eigenvalues of the Laplacian

1) Notes of teachers

2) G. B. Folland, Real Anayisis: modern techniques and their applications, J. Wiley, 1999

3) H.Brezis, Analisi Funzionale, Liguori

4) L.C. Evans, Partial Differential Equations, American Mathematical Society

5) A. Ambrosetti - G. Prodi, A primer of nonlinear analysis, Cambridge University Press, 1993

6) Kolmogorov-Fomin: Elementi di teoria delle funzioni e analisi funzionale