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Dades generals

 

Nom de l'assignatura: AnÓlisi Funcional i Equacions en Derivades Parcials

Codi de l'assignatura: 573765

Curs acadŔmic: 2021-2022

Coordinaciˇ: Joaquin Ortega Cerda

Departament: Departament de MatemÓtiques i InformÓtica

crŔdits: 6

Programa ˙nic: S

 

 

Hores estimades de dedicaciˇ

Hores totals 150

 

Activitats presencials i/o no presencials

60

 

-  Teoria

Presencial

 

30

 

-  TeoricoprÓctica

Presencial

 

30

Treball tutelat/dirigit

20

Aprenentatge aut˛nom

70

 

 

CompetŔncies que es desenvolupen

 

  • Capacity of understanding the concepts and rigorous proofs of fundamental theorems of Functional Analysis and Partial Differential Equations and transverse areas of mathematics.
  • Capacity to apply the results and techniques learned to solve complex problems in different areas of mathematics in academic or professional contexts.
  • Ability to prepare and develop logical-mathematical reasoning and identify errors in incorrect reasoning.
  • Capacity to know how to construct, interpret, analyze and validate mathematical models developed to simulate real situations.
  • Ability to enunciate and to verify statements, and to convey the mathematical knowledge acquired orally or in writing.
  • Capacity to choose and use software tools to address problems related to mathematics.
  • Ability to work in groups.

 

 

 

 

Objectius d'aprenentatge

 

Referits a coneixements

  • To learn the basic results on Banach and Hilbert spaces and operators, with special attention to duality.
  • To know the theory of distributions and Sobolev spaces, mainly in the context of Hilbert spaces.
  • To use the techniques of Functional Analysis in the study of Partial Differential equations.

 

 

Blocs temÓtics

 

1. Hilbert spaces: orthogonality, duality and elementary spectral theory.

2. Banach spaces. Boundedness of linear operators.

3. Sobolev spaces: Regularity and compactness

4. Applications to PDE

 

 

Metodologia i activitats formatives

 

The different teaching blocks will be developed in theoretical classes; For each block, a collection of problems will be distributed and discussed in class; Students will work on all or part of the exercises from the list; After the correction, they will be solved on the blackboard.

 

 

Avaluaciˇ acreditativa dels aprenentatges

 

Grades are awarded based on participation in practical classes and the preparation and presentation of assignments. There will be a longer assignment at the end of the course.

 

Avaluaciˇ ˙nica

For students who opt out of continuous assessment (problem-solving exercises and assignments), a final examination must be completed.

In order to be considered for re-evaluation, the student needs to have at least 3.5/10 points as a final grade.

 

 

Fonts d'informaciˇ bÓsica

Consulteu la disponibilitat a CERCABIB

Llibre

Adams, R. A. Sobolev spaces. Amsterdam : Academic Press, 2003.

Brézis, H. Análisis funcional : teoría y aplicaciones. Madrid : Alianza, 1984.

Brézis, H. Functional analysis, Sobolev spaces and partial differential equations. New York :
Springer, 2011.

Cerdà, J. Introducció a l’anàlisi funcional. Barcelona, Edicions UB, 2005.

Cerdà, J. Linear functional analysis, Providence, R.I. : American Mathematical Society ; Madrid : Real Sociedad Matemática Española, 2010.

Lax, P. Functional analysis. New York : Wiley, 2002.

Maz’ia, V. G. Sobolev spaces. Berlin : Springer, 1985.

Rudin, W. Functional analysis. New York : McGraw-Hill, 1991.

L. Evans, Partial Differential Equations. American Mathematical Society 2010