While it is believed that many particle systems have periodic ground states, there are few rigorous crystallization results in two and more dimensions. In this talk I will show how results by the Hungarian geometer László Fejes Tóth can be used to prove that an idealised block copolymer energy is minimised by the triangular lattice. I will also discuss a numerical method for a broader class of optimal location problems and some conjectures about minimisers in three dimensions. This is joint work with Mark Peletier, Steven Roper and Florian Theil.

# Past Partial Differential Equations Seminar

The Ginzburg-Landau functional serves as a model for the formation of vortices in many physical contexts. The natural gradient flow, the parabolic Ginzburg-Landau equation, converges in the limit of small vortex size and finite number of vortices to a system of ODEs. Passing to the limit of many vortices in this ODE, one can derive a mean field PDE, similar to the passage from point vortex systems to the 2D Euler equations. In the talk, I will present quantitative estimates that allow us to directly connect the parabolic GL equation to the limiting mean field PDE. In contrast to recent work by Serfaty, our work is restricted to a fairly low number of vortices, but can handle vortex sheet initial data in bounded domains. This is joint work with Daniel Spirn (University of Minnesota).

For maps from surfaces there is a close connection between the area of the surface parametrised by the map and its Dirichlet energy and this translates also into a relation for the corresponding critical points. As such, when trying to find minimal surfaces, one route to take is to follow a suitable gradient flow of the Dirichlet energy. In this talk I will introduce such a flow which evolves both a map and a metric on the domain in a way that is designed to change the initial data into a minimal immersions and discuss some question concerning the existence of solutions and their asymptotic behaviour. This is joint work with Peter Topping.

I will discuss a recent joint work with A. Malchiodi (Pisa) and M. Micallef (Warwick) in which we show that not every harmonic map can be approximated by a sequence of $\alpha$-harmonic maps.

The classical Gehring lemma for elliptic equations with measurable coefficients states that an energy solution, which is initially assumed to be $H^1$ - Sobolev regular, is actually in a better Sobolev space space $W^{1,q}$ for some $q>2$. This a consequence of a self-improving property that so-called reverse Hölder inequality implies. In the case of nonlocal equations a self-improving effect appears: Energy solutions are also more differentiable. This is a new, purely nonlocal phenomenon, which is not present in the local case. The proof relies on a nonlocal version of the Gehring lemma involving new exit time and dyadic decomposition arguments. This is a joint work with G. Mingione and Y. Sire.

We study the low-temperature limit in the Landau-de Gennes theory for liquid crystals. We prove that for minimizers for orientable Dirichlet data tend to be almost uniaxial but necessarily contain some biaxiality around the singularities of a limiting harmonic map. In particular we prove that around each defect there must necessarily exist a maximal biaxiality point, a point with a purely uniaxial configuration with a positive order parameter, and a point with a purely uniaxial configuration with a negative order parameter. Estimates for the size of the biaxial cores are also given.

This is joint work with Apala Majumdar and Adriano Pisante.

The massless Maxwell-Klein-Gordon system describes the interaction between an electromagnetic field (Maxwell) and a charged massless scalar field (massless Klein-Gordon, or wave). In this talk, I will present a recent proof, joint with D. Tataru, of global well-posedness and scattering of this system for arbitrary finite energy data in the (4+1)-dimensional Minkowski space, in which the PDE is energy critical.