Unit APPLIED FUNCTIONAL ANALYSIS
- Course
- Mathematics
- Study-unit Code
- A003039
- Curriculum
- Matematica per l'economia e la finanza
- Teacher
- Patrizia Pucci
- Teachers
-
- Patrizia Pucci
- Hours
- 63 ore - Patrizia Pucci
- CFU
- 6
- Course Regulation
- Coorte 2022
- Offered
- 2023/24
- 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
- ENGLISH
- Contents
- The aim of the course is to give basic properties on Applied Functional Analysis (Sobolev spaces. Compact operators. Elliptic PDE's, Evolution equations) and to present problems and applications that come from the physical, biological, chemical and economical phenomena.
- Reference texts
- H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, 2010.
2011, xiv+599 pp.
G. Morosanu, Functional analysis for the applied sciences, Universitext, Springer, Cham, 2019, xii+432 pp. - Educational objectives
- The student should acquire a basic knowledge in Applied Functional Analysis as well as in Sobolev space theory. The course matter is part of the contents of a standard reformed second level course for Italian master degrees in Mathematics. Even the setting is reformed, and the textbooks used are rich of examples and counterexamples, and therefore seem to be optimal to achieve a good understanding of definitions and statements of theorems.
The course aims at analyzing the basic arguments of functional analysis in Banach spaces, at treating so widespread and comprehensive discipline as taught for years at national and international levels. In this sense, the purpose of the course is to make the students able
- To know the main topics of applied functional analysis and how to apply them to the natural sciences,
- To own computational skills to solve various exercises,
- To read and understand texts of Applied Functional Analysis,
- To provide themselves a mathematical proof of simple statements, with strong reasoning skills,
- To communicate in Italian and in English the mathematical knowledge acquired in the course, as well as related issues,
- To work in teams, but also in autonomy.
In summary, he main aim of the course is to understand the application of Applied Functional Analysis to some PDE's. The knowledge acquired will be the one listed in the program. The main ability acquired will be the skill to build a satisfactory theory on existence, uniqueness and continuous dependence on data for some PDE's. The skills listed above are set out in the framework of the professions related to both a traditional mathematician, and a mathematician oriented to technical and/or industrial activities. - Prerequisites
- To better understand the topics covered in the course the student should know the basic subjects of Mathematical Analysis acquired in any Bachelor Degree in Mathematics, Physics and/or Engineering and the main arguments covered in ty he previous course named Functional Analysis. Some elementary knowledge of partial differential equations, usually acquired in basic courses in Mathematical Physics of any Bachelor Degree, is also important. In particular, the course aims at making the student familiar with the theories that play a central role in modern mathematics, such as functional analysis in Sobolev spaces, with its use in applications.
- Teaching methods
- The course is split into traditional lectures, in which several exercises are carried out in class to facilitate the understanding of the course. The essential arguments are summarized in handouts provided by the teacher. The course is divided into 63 hours of theory, together with different examples and counterexamples (almost 20 hours are dedicated to practical exercises). In the tutorial service the students will be followed individually by the teacher.
To better understand the topics covered in the course the student should know the basic concepts of Mathematical Analysis acquired in any Bachelor Degree in Mathematics, Physics and/or Engineering. In particular, the course aims at making the student familiar with the theories that play a central role in modern mathematics, such as functional analysis in Sobolev spaces, with its use in applications.
The lecture attendance, even if not mandatory and compulsory, is strongly recommended for a better understanding of the course. - Other information
- The teacher will distribute educational materials useful for a better understanding of the course, in order to help and to let the students pass easily the exam.
The course is divided in 6 hours per week and its time schedule is available at the web pagehttp://www.dmi.unipg.it/didattica/corsi-di-studio-in-matematica/matematica-magistrale/orario-lezioni
Tutorial service is given in office hours. Customized consulting students, also at the request of the students. - Learning verification modality
- The exam includes a single oral test with the performance of some critical exercises. The oral exam consists of a discussion on three topics one of which divided into several questions and takes about 30 minutes. The oral test is designed to assess the level of knowledge attained by the student on the theoretical contents and on the methodologies of the course (fundamental theorems, definitions, examples and counterexamples). Finally, the oral examination allows the teacher to verify the performance of the student and his/her ability to organize the presentation in autonomy.
The examination timetable is divided into at least 8 exam sessions and the examination dates are available at the web page http://www.dmi.unipg.it/didattica/corsi-di-studio-in-matematica/matematica-magistrale/calendario-esami
For information on support services for students with disabilities and/or DSA visit the page http://www.unipg.it/disabilita-e-dsa - Extended program
- Sobolev Spaces. Lax-Milgram Theorem. Compact operators: definition, properties, adjoint operators, Fredholm alternative, spectrum and spectral decomposition. Elliptic linear problems, existence, uniqueness, multiplicity and regularity. Maximum principles. Eigenfunctions and eigenvalues. The energy method for heat and wave equations.
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