The exercises relative to the class of the week will appear on the day of the class; they will be discussed during the exercise class of the following week (to give time to follow the class and revise the material); the solutions will appear on Friday evening every week.

Each week, one (or, in some cases, 2) student can volunteer to present a problem if he wants to. We encourage every student to present at least one problem during the whole semester. 

Lectures: Thursday, 8.15 – 10.00, GR A3 30
Exercises: Thursday, 10.15 – 12.00, GR A3 30

Course content   

After recalling classical results of topology and functional analysis (the three Banach theorems), we will define the notions of weak topology and weak * topology, and introduce the notions of reflexive and separable spaces.

Following this introductory section, we will introduce (locally convex) topological vector spaces that will be needed to define topologies on the space of distributions—linear forms on the space of smooth function with compact support. Indeed, the dual topology on distributions is not metrisable. We will afterwards introduce all basic notions related to distributions: product, division, boundedness, convergence, completeness, differentiation, localisation, support before moving to the last two main topics: convolution and Fourier transform. Distributions generalise functions, and can be used to give a meaning to partial differential equations (PDEs) that would classically make sense for smooth of regular enough functions, although the only physical information available is typically a weak control—or the kinetic energy for example (the gradient of the function is squared-integrable). Furthermore, the operation of differentiation of those generalised functions is continuous, which allows one to pass to the limit in PDEs under weak hypotheses.

The convolution of distributions will extend the familiar notion for $L^p$ functions, and permit one to give an alternative formulation of linear partial differential equations, and give a precise sense to the notion of fundamental solution. Finally, the Fourier transform, that can only be defined on a subspace of distributions—known as the Schwartz space, or space of tempered distributions—will be introduced, and links to PDEs emphasised. 

The next part of the course will be about Sobolev spaces—spaces of distributions whose derivatives belong to some $L^p$ spaces. Contrary to the space of distributions, Sobolev spaces are Banach spaces, and one can generally give a meaning to most PDEs within this framework where existence theorems are relatively easier, whilst regularity theory is more challenging. The main focus will be on Sobolev embedding theorems—if the gradient of a distribution is an $L^p$ function, then the distribution is itself a $L^q$ function, where the parameter $q$ depends on $p$ and the ambient dimension $d$—and Sobolev inequalities, and their generalisations.

In the penultimate section of the course, we will introduce interpolation, starting by Sobolev spaces of fractional exponents modelled on the Hilbert space $L^2$, before moving to more abstract notions.

In the last part of the lecture, we will give new applications to PDEs, and if time remains, introduce the notions of BMO, Lorentz and Orlicz spaces, that happen to be of crucial importance in a wide variety of applications, and permit to precise the Sobolev embedding theorems in the limiting cases.