Self-improving properties with applications to PDEs

In this project, I will specifically focus on the self-improving property of the quasi-minimisers of certain functionals. By the self-improving property of weak reverse Holder inequalities, we have a higher regularity of a quasi-minimiser of the p-variational integral. The goal of this project is to explore whether we can obtain an improvement of the quasi-minimising property. In other words, it is to see whether a quasi-minimiser of the p-variational integral is a quasi-minimiser of the q-variational integral for some q>p.

As a starting point, I am going to work on some simple domains like cubes and balls, for which we consider cubic and spherical quasi-minimisers. The second step is to extend it to general domains.

If the results above could be achieved, I am going to further investigate the size of the potential improvement and how the corresponding quasi-minimising constant varies according to the exponent q.

One possible application of this work is about PDEs, since the solutions to PDEs under proper conditions are quasi-minimisers.

After finishing all the steps above, I am going to do some further work in this field, considering problems about general elliptic regularity.

To do this project, some preparation and a plan are necessary before I really work on it. For this project, I need to read literatures on self-improving properties, capacities, p-Laplace equations, etc., to learn some helpful tools and techniques. This has been partially done. Besides, it is important and also difficult to obtain a rough idea and to develop a practical strategy to solve the problem. In this project, the plan is to apply the minimising property of p-harmonic functions to describe quasi-minimisers and then to have a further investigation. The following step will be to work everything out concretely, which might be time-consuming and involve many details.

The work, after being completed, will be published in international peer-viewed journals.

This project falls within the EPRSC mathematical analysis research area.

The core purpose of this proposed project is to establish a world-class doctoral training centre of excellence in PDEs, with a vibrant and stimulating training/research environment and a dynamic and sustainable doctoral training base, to train a new generation of first-rate UK leaders in academic and industrial research in PDEs and their analysis/applications with a portfolio of transferable, interdisciplinary, organisational and ethical skills to interact with industry and society and to help drive scientific advances for the next fifty years.

The proposed CDT cohort training programme will equip students with a broad and deep expertise in the analysis and applications of PDEs and related areas of Core Mathematics and its Interfaces, and will engage with other areas of the mathematical sciences and academic/non-academic users. In current practice, students often narrow down in an abrupt, stepwise fashion to a specific research project, leaving them with an incomplete, imbalanced perception of the research purpose, context, and value. Such a cohort training, beyond the traditional training practice, will provide students with new opportunities not only to learn many diverse aspects of PDEs and related analysis/applications through group-work and mutual support, but also to develop their ability to engage with a range of communities and to understand and communicate with experts in diverse topics.

We envisage impacts of several different kinds resulting from the project:

1. Creation of a world-class doctoral training centre of excellence in PDEs, which, because of the prevalence of PDEs in science and engineering, impinges on the majority of the EPSRC CDT call priority areas;

2. Involvement of more than 50 DPhil students, who will be trained by the CDT in PDEs and other related CDT priority areas and will go on to have careers in academia or applied sectors in industry and government;

3. Advancement of the field of PDEs, Analysis, and related Core Mathematics and its Interfaces within the UK, as well as other fields such as the global challenge areas of energy and climate change, by delivering a new generation of home-grown research leaders in the priority areas, which will also give the UK an improved standing (reputation and influence) within the global mathematical research community;

4. Reinforcement of the UK's knowledge-based economy, by providing users in commerce, industry, and government agencies that rely on an understanding of PDEs and related analysis with a new generation of problem-solvers and innovators;

5. Creation of a lasting web-based archive of lectures on major contributions to the field and delivery of web-based teaching materials with benefit to the wider community;

6. Development and application of new mathematical research in PDEs and related Core Mathematics and its Interfaces, which will be published in journal articles, conference proceedings, and arXiv articles, and exposed in seminar and conference talks, so that mathematicians and the wider scientific/industrial community can learn about and use the results, and pursue further research.