Although we do not yet know how long the social distancing related to the Covid-19 pandemic will last, and regardless of the changes that had to be made in the evaluation of the June 2020 session in relation to what is provided for in this learning unit description, new learnig unit evaluation methods may still be adopted by the teachers; details of these methods have been - or will be - communicated to the students by the teachers, as soon as possible.
LMAT1321 Analyse fonctionnelle et équations aux dérivées partielles (or an introductory course on Functional Analysis),
LMAT1322 Théorie de la mesure (or an introductory course on measure theory and the Lebesgue integral).
At the end of this learning unit, the student is able to :
Contribution of the course to learning outcomes in the Master in Mathematics programme. By the end of this activity, students will have made progress in:
- Choose and use calculation tools to solve mathematical problems.
- Identify the fundamental concepts of important current mathematical theories.
- Establish the main connections between these theories, analyse them and explain them through the use of examples.
- Identify, by use of the abstract and experimental approach specific to the exact sciences, the unifying features of different situations and experiments in mathematics or in closely related fields.
- Show evidence of abstract thinking and of a critical spirit.
- Argue within the context of the axiomatic method.
- Construct and draw up a proof independently, clearly and rigorously.
- Recognise the key arguments and the structure of a proof.
- Evaluate the rigour of a mathematical or logical argument and identify any possible flaws in it.
- Distinguish between the intuition and the validity of a result and the different levels of rigorous understanding of this same result.
- Write a mathematical text according to the conventions of the discipline.
- Find sources in the mathematical literature and assess their relevance.
- Correctly locate an advanced mathematical text in relation to knowledge acquired.
- Ask relevant and lucid questions on an advanced mathematical topic in an independent manner.
Learning outcomes specific to the course. By the end of this activity, students will be able to:
- State, prove and illustrate propositions concerning properties of solutions of partial differential equations, and also the existence and uniqueness of such solutions.
- Propose one or several strategies to establish the existence of solutions.
- Apply tools from real analysis to solve a problem.
- Manipulate notions from advanced analysis.
- Contextualize mathematical tools in their historical setting and understand how they evolved.
The contribution of this Teaching Unit to the development and command of the skills and learning outcomes of the programme(s) can be accessed at the end of this sheet, in the section entitled “Programmes/courses offering this Teaching Unit”.
Harmonic functions: Mean value property, regularity, maximum principle
Harnack inequality, Liouville Theorem
Gauss-Green formulas, fundamental solution, distributions, Green's function
Sobolev spaces, elliptic boundary value problems
Heat equation: Fundamental solution, maximum principle, regularity
Wave equation: Explicit solution
- reproduce material, especially definitions, theorems, proofs and examples
- demonstrate a certain mastery of the available tools
- explain the limits of a method or a tool
Assessment will be on the basis of
- knowledge, understanding and application of the different mathematical objects and methods from the course
- precision of calculations
- rigour of arguments, proofs and reasons
- quality of presentation of answers
- Lawrence C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, AMS, 2010.
- Augusto C. Ponce, Elliptic PDEs, Measures and Capacities, EMS Tracts in Mathematics, vol. 23, European Mathematical Society (EMS), Zürich, 2016.
- matériel sur moodle