Project IV (MATH4072) 2025/26

What's beyond Lebesgue spaces?

Katie Gittins & Wilhelm Klingenberg

Description

The fundamental theory of Lebesgue introduced a new perspective on integration which profoundly impacted mathematics. Integration is intimately linked to differentiation so we could ask: is there some sort of generalised notion of differentiation that makes sense in the setting of Lebesgue spaces?

Good news: there is! Motivation and intuition for what it means to be "weakly differentiable" are plentiful, and perhaps surprisingly, lead to the same idea. Here are a few fascinating paths that motivate such a notion:

  • Approximation - Approximation of a function by a sufficiently nice function (in an appropriate norm) allows us to uncover rich analytic properties of the space of functions we're exploring. What do we get when we try to mix classic differentiation with the notions of the Lebesgue spaces?
  • Existence of solutions to PDEs - Moving away from classical (that is, differentiable) solutions of PDEs and taking inspiration from Lebesgue theory, consider the equation \[ \left\{ \begin{array}{ll} -\Delta u(x)=f\left(x\right), & x\in \Omega,\\ u(x)=0, & x\in \partial \Omega, \end{array} \right. \] where \(\Omega \subset \mathbb{R}^2\) is your favourite planar domain and \(f \in L^2(\Omega)\). By multiplying it with a suitably nice function \(\phi\) and integrating by parts we find that we must have \[ \int_{\Omega} \nabla u(x) \cdot \nabla \phi(x) \,dx = \int_{\Omega}f(x)\phi(x) \, dx.\] This expression even makes sense in a far wider class of functions than continuously differentiable functions, giving us a different way to define "\(\nabla u\)". You might remember (from Analysis III) that considering this framework allows us to prove existence of the solution to the above equation in a suitable Hilbert space via the Riesz Representation theorem. But, what is this Hilbert space?
  • Optimisation problems - A fundamental principle in our understanding of the natural world is that things tend to happen in a way that takes the least action . Mathematically, this means that a solution to a particular problem usually minimises an appropriate physical functional such as an energy. An example, associated to understanding the frequencies that a drum \(\Omega\) vibrates at, is expressed via the eigenvalue problem \[ \left\{ \begin{array}{ll} -\Delta u(x)=\lambda u(x), & x\in \Omega,\\ u(x)=0, & x\in \partial \Omega, \end{array} \right.\] and relates to the so-called Rayleigh quotient \[ R[u] = \frac{\int_{\Omega}\left | \nabla u(x)\right|^2dx}{\int_{\Omega}\left | u(x)\right|^2dx}. \] But, what would be the right space over which the functional \(R[u]\) is guaranteed to attain a minimum?

    • The function spaces where all this action takes place are the celebrated Sobolev Spaces. Not only do they allow us to glean information about the directions we mentioned above, but they also enjoy a wealth of beautiful properties!

    Our first goal in this project will be to build a solid foundation in the theory of Sobolev spaces including the following topics:

    • The definition and properties of weak derivatives and Sobolev spaces.
    • Approximation by smooth functions.
    • Sobolev embeddings and inequalities.

    After laying these foundations together, you would be free to choose your preferred direction in which to continue your investigation. Some possible directions include:

    • Extensions and restrictions (traces).
    • Important geometric and functional inequalities such as Poincaré's inequality and Hardy's inequality.
    • Differentiability almost everywhere.
    • The study of the vibrational frequencies of a drum with fixed boundary.
    • The connection between Sobolev spaces and the Fourier transform.
    • Regularity theory for elliptic equations.

    Prior and Companion Modules

    Essential Prior Modules : Analysis III.

    Depending on your preferred direction of the project, the module Partial Differential Equations III could be useful but is not essential.

    The module Functional Analysis and Applications IV could be taken as a companion module as it complements some elements of this project.

    Resources

    • R.A. Adams, J.J.F. Fournier, Sobolev Spaces, Elsevier, 2003.
    • L.C. Evans, Partial Differential Equations, AMS, 2010.
    • L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1991.
    • Richard S. Laugesen , Linear Analysis and Partial Differential Equations.

    If you would like more information about this project, then please feel free to contact us via email: K Gittins and W Klingenberg