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
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).
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… Read More »
We show existence by showing convergence of a suitable finite element scheme. This is joint work with G. Gruen and H. Grillmeyer.
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).
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… Read More »
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… Read More »
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… Read More »
TBC