# PhD Studentship: New Regularity Methods in Nonlinear Elasticity

### University of Surrey

Supervisor: Dr Jon Bevan

Funding: Competition Funded Project (Students Worldwide)

Project description: This project would be of interest to any mathematician whose interests already include analysis and PDEs, and who would be prepared to work on new and interesting problems that arise in nonlinear elasticity theory and/or the calculus of variations. No prior experience in nonlinear elasticity is required for this: the successful candidate will be given the required training.

A basic tenet of the theory is that the physical configurations of an elastic material (e.g. rubber) that we can expect to observe are those which minimize an elastic energy functional. When expressed as an integral, as is the case here, the task of minimizing the elastic energy then becomes a problem in the vectorial calculus of variations with various technically challenging features. A major open question of some 40 years’ standing in nonlinear elasticity theory is to discover how smooth the minimizers of elastic energies should be. Recent progress has been made towards solving this problem in [Bevan, arXiv 1509.08245] with the discovery and characterization of a new property of elastic deformations called `positive twist’. Local minimizers with this property are automatically more regular than a general admissible map, which is the goal of regularity theory and is in accordance with our physical intuition:  we would, generally, expect to find that, under the right conditions, energy minimizers are indeed smooth. The theory set out in [Bevan, arXiv 1509.08245] represents the first steps in a new direction, but there are still many exciting things to discover about the property of positive twist and what it can tell us about energy minimizers. For example, in the case of planar maps, what are the connections between positive twist and `vorticity’ of the kind encountered in fluid mechanics, when suitably interpreted? Or, in higher dimensions, what is the natural analogue of positive twist, and can it be characterized in terms of `star-shapedness’, as is the case with the two-dimensional version? These and other questions could form the centrepiece of a cutting-edge PhD project in applied analysis/PDE theory/nonlinear elasticity.

The thesis supervisor, JB, is a Senior Lecturer in Mathematics and an established applied analyst with over 15 years’ experience in the field.

References:  [Bevan, arXiv 1509.08245] J. Bevan. A condition for the Hölder regularity of local minimizers of a nonlinear elastic energy in two dimensions. Preprint, 2016.

Funding: The Faculty has a fully funded PhD studentship for UK and EU nationals who can demonstrate the appropriate residency requirements. This studentship will include the tuition fees and a tax-free stipend. The Department has also a few scholarships for partial funding for overseas fees. However, funding for overseas students is limited and overseas students are encouraged to find suitable funding themselves.

Application enquiries: Dr Matt Turner- Maths-phd@surrey.ac.uk

Subject areas: Applied Mathematics, Mathematics