Subject: ADVANCED ANALYSIS (A.A. 2020/2021)
Unit Analisi superiore
Advanced Theoretical Studies (lesson)
The course provides the student with the elements of Functional Analysis, with special emphasis on Banach spaces and Fourier Transform. Moreover, the basic knowledge of the thoery of lienar Partial Differential Equations is given.
Limits, continuity, differentiantibility for real and vector valued functions of several real variables, Lebesgue integral
Basic theory of topological vector spaces. Linear operators. Hahn-Banach theorem. Baire theorem. Dual spaces, weak topology. Hilbert spaces. L^p spaces. Convolutions and mollifiers. Weak derivatives. Sobolev spaces. Embedding theorems.
Fourier transform. Schwartz space. H^s spaces. Trace of functions belonging to H^s.
Applications to boundary value problems for elliptic equations.
Fundamental solution of the heat equation. Existence and uniqueness results for the Cauchy problem for the heat equation. Characterstic method for I order Partial Differential Equations and for Hyperbolic Partial Differential Equations.
Lectures, synchronous remote delivery will be guaranteed. Based on the evolution of the health emergency COVID19, the option of the theaching activity in presence will be evaluated.
Oral exams. The exams will be carried out either in presence or in remote, depending on the evolution of the COVID19 situation.
1. Knowledge and understanding
Through lectures and individual study, knowledge of the main concepts in Functional Analysis, in particular for Hilbert and Banach space and for the Fourier transform.
2. Applying knowledge and understanding
Through classroom exercises, support activities and individual work, ability to model and solve mathematical problems by analytical tools.
3. Making judgements:
Attitude towards a methodological approach that leads to verify claims and methods through rigorous arguments.
Ability to self-assessment of their skills and abilities.
4. Communication skills:
Ability to deal with a dialectical discourse in a timely and consistent way, arguing with precision.
Lasting acquisition of mathematical knowledge to be used at any time of their cultural journey.
5. Learning skills:
Attitude towards a methodological approach that leads to an improvement in the method of study in order to deepen learning capacity.
Durante il corso verranno date dispense e indicazioni bibliografiche.