Two of the fundamental issues in the analysis of generalised solutions for nonlinear PDEs are the weak rigidity/continuity of nonlinear PDEs and the compactness/convergence of approximate/multiscale solutions. In this talk, we will discuss some recent developments on these issues for several important classes of nonlinear PDEs

# Author Archives: Service Account

## The $p$-Hunter-Saxton equation

## Carlo Mercuri – A compactness result for Schrödinger-Poisson systems.

I will present a compactness result for certain sequences of approximated critical points of functionals (Palais-Smale sequences) related to a class of Schrödinger-Poisson systems. Applications will be discussed in relation to the minimax approach for finding positive solutions to these systems. This is a joint work with Megan Tyler (PhD student, Swansea University).

## Elaine Crooks – Invasion speeds in a competition-diffusion model with mutation.

We consider a reaction-diffusion system modelling the growth, dispersal and mutation of two phenotypes. This model was proposed in by Elliott and Cornell (2012), who presented evidence that for a class of dispersal and growth coefficients and a small mutation rate, the two phenotypes spread into the unstable extinction state at a single speed that is faster than either phenotype would spread in the absence of mutation. After first showing that, under reasonable conditions on the mutation and competition parameters, the spreading speed of the two phenotypes is indeed determined by the linearisation about the extinction state, we prove that the spreading speed is a non-increasing function of the mutation rate (implying that greater mixing between phenotypes leads to slower propagation), determine the ratio at which the phenotypes occur in the leading edge in the limit of vanishing mutation, and discuss the effect of trade-offs between dispersal and growth on the spreading speed of the phenotypes. This talk is based on joint work with Luca Börger and Aled Morris (Swansea).

## Nicolas Dirr – Existence of solutions and convergence of a finite-element scheme for a stochastic pororus-medium equation with multiplicative noise in divergence form.

We show existence by showing convergence of a suitable finite element scheme. This is joint work with G. Gruen and H. Grillmeyer.

## PDEs and probability – Horatio Boedihardjo

We will discuss the classical relationships between probability and PDEs, as well as some recent developments. In particular, we will explain our ongoing study of an eigenvalue problem associated with a linear matrix-valued elliptic PDE from probability theory. Joint work with Ni Hao (UCL).

## Stochastic homogenisation of high-contrast media – Mikhail Cherdantsev

Using a suitable stochastic version of the compactness argument of V. V. Zhikov, we develop a probabilistic framework for the analysis of heterogeneous media with high contrast. We show that an appropriately defined multiscale limit of the field in the original medium satisfies a system of equations corresponding to the coupled “macroscopic” and “microscopic” components of the field, giving rise to an analogue of the “Zhikov function”, which represents the effective dispersion of the medium. We demonstrate that, under some lenient conditions within the new framework, the spectra of the original problems converge to the spectrum of their homogenisation limit.

## On the existence and uniqueness of vectorial absolute minimisers in Calculus of Variations in L-infinity – Nikos Katzourakis

Calculus of Variations in the space L-infinity has a relatively short history in Analysis. The scalar-valued theory was pioneered by the Swedish mathematician Gunnar Aronsson in the 1960s and since then has developed enormously. The general vector-valued case delayed a lot to be developed and its systematic development began in the 2010s. One of the most fundamental problems in the area which was open until today (and has been attempted by many researchers) concerned that of the title. In this talk I will discuss the first result in this direction, which is based on joint work with my research associate Giles Shaw.

## Uniqueness of minimisers of Ginzburg-Landau functionals – Luc Nguyen

We provide necessary and sufficient conditions for the uniqueness of minimisers of the Ginzburg-Landau functional for $\RR^n$-valued maps under suitable convexity assumption on the potential and for $H^{1/2} \cap L^\infty$ boundary data that is non-negative in a fixed direction $e\in \SSphere^{n-1}$. Furthermore, we show that, when minimisers are non-unique, the set of minimisers is invariant under appropriate orthogonal transformations of $\RR^n$. As an application, we obtain symmetry results for minimisers corresponding to symmetric Dirichlet boundary data. We also prove corresponding results for harmonic maps. Joint work with R. Ignat, V. Slastikov and A. Zarnescu.