Teaching plan for the course unit
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General information

 

Course unit name: Functional Analysis and Partial Differential Equations

Course unit code: 573765

Academic year: 2023-2024

Coordinator: Joaquin Ortega Cerda

Department: Department of Mathematics and Computer Science

Credits: 6

Single program: S

Estimated learning time

Total number of hours 150

 

Face-to-face and/or online activities

60

	-  Lecture	Face-to-face		30
	-  Lecture with practical component	Face-to-face		30

Supervised project

20

Independent learning

70

Competences / Learning outcomes to be gained during study

 

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.

Learning objectives

Referring to knowledge 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.

 

 

Teaching blocks

 

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

 

 

Official assessment of learning outcomes

 

Grades are calculated based on a grade (P) that assesses participation in practical classes and the preparation and presentation of problems. There will also be a test at the end of the course (F). The final grade is max(F, 0*4P+0.6F).   Examination-based assessment For students who opt out the continuous assessment, a final exam will have to be taken. To be considered for re-evaluation, the student must have at least 3.5/10 points as a final grade.

Reading and study resources

Check availability in Cercabib

Book

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