The calculus of variations is, roughly speaking, the mathematical theory of how to minimise or maximise magnitudes. Variational problems from science typically require minimisation of an average magnitude over a space-time region depending on unknown quantities. But not all variational problems pertain to an average. For shape or design optimisation problems, for example, it is a maximum that needs to be minimised instead, which leads to minimax problems. At present, no general mathematical theory exists for minimax problems that involve several unknown quantities simultaneously. Variational problems in L∞ may not typically arise as fundamental physical laws, but they appear in design and optimisation problems, especially when it is does not suffice to prescribe and minimise just the average of a quantity. Apart from being of fundamental analytical interest, they are very useful, and sometimes pivotal, for applications. Especially when the functional represents a cost or an error, minimising the maximum rather than an Lp-integral will improve the performance of many specific models. Minimisation of the maximum guarantees uniform smallness, as spike deviations, even of small Lp-norm, are excluded from the outset. The analogues of the Euler-Lagrage equations arising for L∞ functional are in general fully nonlinear and potentially with discontinuous coefficients, so novel theories of generalised solutinos are required for their rigorous study.
The purpose of this project is to develop new mathematical theories with a novel approach, building on the work done by the supervisor and collaborators/students in the past 10 years.
There are many open problems in this rapidly developing area, so the successful applicant might choose their own from a broad pallette of possibilities, after mutual agreement with the supervisor.
Eligibility: A BSc (2:1 or above) in Mathematics, preferably with an MMath or MSc in pure mathematics or mathematical analysis. Good knowledge of measure theory and functional analysis is essential, or at least a strong desire to learn these.