Unit APPLIED FUNCTIONL ANALYSIS
- Course
- Mathematics
- Study-unit Code
- 55A00077
- Curriculum
- Matematica per l'economia e la finanza
- Teacher
- Enzo Vitillaro
- Teachers
-
- Enzo Vitillaro
- Hours
- 63 ore - Enzo Vitillaro
- CFU
- 6
- Course Regulation
- Coorte 2020
- Offered
- 2021/22
- Learning activities
- Caratterizzante
- Area
- Formazione teorica avanzata
- Academic discipline
- MAT/05
- Type of study-unit
- Opzionale (Optional)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- Sobolev spaces. Compact operators. Elliptic PDE's- Bochner ans Sobolev spaces of vector valued functions. Evolution equations.
- Reference texts
- 1. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, 2010.
2. L.Evans Partial Differential Equations. Graduate Studies in Mathematics,19, American Mathematical Society 1998
3. Lecture Notes by the teacher. - Educational objectives
- The main aim of the COurse is to understand the application of Linear Functional
Analysis to linear P.D.E.'s.
The knowledge acquired will be the one listed in the program.
The main ability acquired will be the ability to build a satisfactory theory on existence, uniqueness and continous dependence on the data for a linear PDE. - Prerequisites
- In order to be able to understand and apply the majority of the techniques described in the course the content of the courses: ANALISI MATEMATICA I,II,III,IV and V are mandatoty, as a basic knowledge of Linear Algebra.
Some elementary knowledge of partial differential equations, usually learned in basic courses in Mathematical Physics, is also important. - Teaching methods
- Face to face.
- Other information
- Depending on the student's preferences, the language used during the course will be Italian or English. Moreover, exam will be in Italian or in English depending on the student preference.
- Learning verification modality
- Oral exam, takin in average 30 minutes, on all the arguements treated, to show up: how much the student understood the theory and how deeply. Moreover the ogranzing, tehnical and expository skills of the student will be investigated.
- Extended program
- Sobolev Spaces. Lax-Milgram Theorem. Compact operators: definition,
properties, adjoint operator, Fredholm alternative, spectrum and
spectral decomposition. Elliptic linear problems, existence,
uniqueness, multiplicity and regularity. Maximum principles.
Eigenfunctions and eigenvalues. Function spaces for Banach-valued
functions. The energy method for heat and wave equations.