About the Project
This scholarship is funded by Swansea University and Université Grenoble Alpes.
Start date: 1 October 2019
Subject areas: Mathematical analysis; Eigenvalue optimization; Nonlinear partial differential equations
Project description and aims:
This scholarship is for a joint Swansea-Grenoble PhD. During the candidature the successful applicant will spend time at the College of Science, Swansea University and the Institut Fourier, Université Grenoble-Alpes, with a minimum requirement of one year spent in each institution.
Originally posed by Rayleigh to study the lowest pitch problem, which seeks a drum of given area that produces the lowest possible note, optimization questions for eigenvalues of second order elliptic operators and shape optimization problems have been widely studied in mathematics, and lie at the intersection of areas such as partial differential equations (PDEs), calculus of variations, and geometry. In parallel, principal eigenvalues of second order elliptic operators play a central role in the study of qualitative and quantitative questions for reaction-diffusion PDEs, often with motivation from, and implications for, applications. Examples of the role played by principal eigenvalues in reaction-diffusion problems include determining the stability of steady states on bounded domains with various boundary conditions, which has applications to the long-time persistence, extinction or coexistence of species, and characterising the minimal speeds of families of travelling waves, which often can be shown to provide quantitative information about the speed of spread of a population.
The aim of this project is to explore new optimization results for principal eigenvalues of elliptic operators and their links to qualitative and quantitative properties of nonlinear partial differential equations, with a focus on non-symmetric operators and their applications to PDEs. Recent advances in the study of eigenvalue shape optimization for non-symmetric elliptic operators and the role of nonlinear convective effects on the dynamics of reaction-diffusion equations open the door to interesting possible extensions and the potential for a fruitful interplay in light of the non-symmetric linearised operators that arise in the study of reaction-diffusion problems with convection.
The two themes of the project, on eigenvalue optimization and dynamics of reaction-diffusion problems, are expected to cross-fertilise and be conducted primarily in Grenoble and Swansea respectively, taking advantage of the complementary expertise in the supervisory team. It is anticipated that the first 18 months of the project will be spent in Grenoble, focussing on eigenvalue optimisation, followed by 18 months in Swansea, where the earlier work on eigenvalue optimisation will feed into research on applications to dynamics of reaction-diffusion equations. Throughout the project, regular discussions will take place with supervisor(s) not present via videoconference, and it is expected that the whole project team will meet in person once per year.
Eligibility
As this is a joint degree, applicants must meet the entry/funder requirements of both universities: a recognised master’s degree in Mathematics and an appropriate English language qualification (https://www.swansea.ac.uk/international-students/requirements/english-requirements/).
A strong background in mathematical analysis is required.
Due to funding restrictions, this scholarship is open to UK/EU candidates only.
Funding Notes
This scholarship covers the full cost of UK/EU tuition fees (50% by Swansea University, 50% by Université Grenoble Alpes) and an annual stipend of £15,009 reviewed every year.
Additional funding is available from Swansea University to assist with travel, accommodation and immersive training experiences.