 Teaching plan for the course unit

 Close  General information

Course unit name: Functional Analysis and Partial Differential Equations

Course unit code: 568175

Coordinator: María Jesús Carro Rossell

Department: Department of Mathematics and Computer Science

Credits: 6

Single program: S

 Estimated learning time Total number of hours 150

 Face-to-face learning activities 60
 -  Lecture 30 -  Lecture with practical component 30
 Supervised project 20
 Independent learning 70

 Competences 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 Ordinary and Partial Differential equations.

 Teaching blocks

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

2. Banach and Frechet spaces. Boundedness of linear operators.

3. Fundamental theorems of Functional Analysis

*  Fundamental theorem of Functional Analysis: Baire, Open mapping, Closed graph, Uniform boundedness principle and Hahn-Banach theorems.

4. Distribution theory.

*   Weak derivatives, convolution, fundamental solutions, temperate distributions and  Fourier transform.

5. Sobolev spaces: Regularity and compactness

6. Applications to PDE

*  Existence of the fundamental solution, Regularity of the solutions, Application of the spectral theory.

 Teaching methods and general organization

 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

 Official assessment of learning outcomes

 This evaluation has 2 parts:Activity 1: Resolution of exercises and other assignments given in class.Activity 2: Written exams.The final grade will be obtained by averaging 20% of  activity 1,  30%   of partial  exams and 50% of the final exam (a minimum of 4/10 points are required on each activity).   Examination-based assessment Final exam with theoretical and practical questions.In order to be considered for re-evaluation, the student needs to have at least 3.5/10 points as a final grade.