Education
Experiences
Awards
School PGR Researcher of the Year
Also nominated for a Vice Chancellor's Award.
EPSRC Studentship
Topics in the multi-dimensional calculus of variations.
Stephen Gourley Prize.
For an excellent final year project
Third Year Departmental Prize
For the best performance in Level 6.
Second Year Departmental Prize
For the best performance in Level 5.
LMS Undergraudate Research Bursary with department matched funding
Bifurcations and dynamics in circle maps derived from 'threshold' dynamical systems
First Year Departmental Prize
For the best performance in Level 4.
Roles
Student Conference Committee Member
PGR Student Representative
Maths Student Mentor Scheme Coordinator
Note Taker
Vice President of MathSoc
Last publications
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Conditions for the Non-Negativity of Excess Functionals in Constrained Variational Problems
Authors: Elliott Farrall
PhD Thesis • September 2025
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Topics in Multi-Dimensional Calculus of Variations
Authors: Elliott Farrall
PhD Confirmation Report • January 2023
Last presentations
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Minimisation of the Dirichlet Energy amongst a class of constrained maps
Introduction to my Research
University of Surrey • December 2024
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Minimisation of a Constrained Energy Functional
Highliights of 2024
University of Surrey • December 2024
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Minimisation of the Dirichlet Energy amongst a class of constrained maps
Analysis Seminar
University of Bath • December 2024
In this talk, we will consider the problem of minimising the Dirichlet energy subject to a pointwise Jacobian constraint with Dirichlet boundary condition. Both the Dirichlet energy and pointwise Jacobian constraints appear in a wide range of physical problems, such as elasticity, which we will briefly discuss. This talk will focus on one particular method for showing that u is a minimiser, which makes use of a so-called excess functional. This functional is interesting in its own right as it serves as an example of a simple (but non-convex) functional that is not always bounded below (depending on the choice of p) and so the direct method may not be of use.
The bulk of the talk will focus mainly on results relating to the excess functional, investigating qualitative changes that occur as p is varied. In particular, we will discuss methods for establishing sucient and necessary conditions for the excess to be bounded below. We will then apply these methods to a particular one parameter family of pressure functions, deriving sucient and necessary bounds on the parameter. Time allowing, we will then derive the maps u, associated with these pressure functions p, that minimise the Dirichlet energy subject to a pointwise Jacobian constraint.
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Wirtz Pumps and Infinite Dimensions
Amazing Maths Seminar
University of Surrey • October 2024
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Mean Hadamard Inequalities & Elasticity
Junior Analysis Seminar
Imperial College London • March 2024
In this talk, I shall discuss a class of functional inequalities known as mean Hadamard inequalities. These inequalities generalise (in some sense) the standard Hadamard inequality. I will show how these inequalities relate to problems in elasticity, in particular, minimisation of the Dirichlet energy subject to a conservation of mass constraint. I will also take some time to discuss some various notions of convexity and how this motivates the search for new techniques to handle these minimisation problems.
Although this talk will mainly be an introduction to the problem of finding mean Hadamard inequalities, I will take some time to go over some results in this area. More specifically, I will show some sufficient conditions that can be derived analytically using Fourier decomposition and weighted Poincaré inequalities and necessary conditions that use novel constructions of solutions to partial differential inclusions and numerical integration.
For further details, see https://imperial.ac.uk/events/174575/elliot-sullinge-farrall-surrey-tbc
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Optimisation of a 1D Family of Polyconvex Functionals
Singularities and Patterns in Evolution Equations
University of Bath • September 2023
In the study of nonlinear elasticity, we often study minimisers of the Dirichlet energy functional to understand how a material will deform in a given situation. Without further conditions, the existence of minimisers is readily established via the direct method. However, complications may arise when the system is given a constraint on the Jacobian. For example, we may specify that we must have unit Jacobian almost everywhere, often referred to as ’incompressibility’.
To incorporate this constraint, one often constructs a new functional (as in the method of Lagrange multipliers) that we will call an ’excess’ functional. To establish the existence of minimisers of the Dirichlet energy, we then need to show non-negativity of the excess functional. Due to how they are constructed, excess functionals are not convex (like the Dirichlet energy) but are polyconvex. The direct method can be applied to polyconvex functionals but much stronger growth/coercivity conditions must be imposed to gurantee the existence of a minimiser, which the excess functionals will not necessarily satisfy. This motivates the usage of other techniques to study these excess functionals.
We now turn our attention to a parameterised 1D family of polyconvex functionals, each of which takes the form of an excess functional. We shall establish necessary and sufficient conditions on the parameter for the excess functional to be bounded below using a combination of analytic and numerical techniques including Fourier expansions, Poincar´e-type inequalities and fractal solutions to partial differential inclusions.
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Optimisation of a 1D Family of Polyconvex Functionals
SIAM Student Chapter Conference
University of Oxford • June 2023
In the study of nonlinear elasticity, we often use the direct method in calculus of variations to establish to existence of minimisers of convex functionals that satisfy reasonable growth (coercivity) conditions. A typical example would be the Dirichlet energy functional under suitable boundary conditions. If we wish to optimise subject to an incompressibility constraint then we can use the method of Lagrange multipliers but the resulting augmented functional is now only polyconvex. To apply the direct method to polyconvex functionals we need to impose much stronger growth conditions that do not apply to the augmented functional.
We now turn our attention to a parameterised 1D family of polyconvex functionals that are related to the augmented functional. We shall establish necessary and sufficient conditions on the parameter for a minimiser of the functional to exist using a combination of analytic and numerical techniques including Fourier expansions, Poincaré-type inequalities and fractal solutions to partial differential inclusions.
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Topology in Analysis
PGR Student Seminar
University of Surrey • May 2023
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Optimisation of a 1D Family of Polyconvex Functionals
Internal Student Conference
University of Surrey • March 2023
The calculus of variations is an area of analysis that predominantly focuses on the optimisation of functionals (functions that map functions to scalars) by considering the effects of making small changes (variations) to functions. This leads to much more complexity than that of the ’standard’ calculus, where we can simply just find zeros of derivatives. In the calculus of variations, great care has to be taken regarding the spaces of functions that we consider.
We will begin this talk with an exploration of some of the key concepts in the calculus of variations, including variational methods and the direct method. We will establish some definitions of convexity that are commonly used in the field and discuss their relevance to the direct method. We then move onto more current developments in the area of optimising polyconvex functionals.
Such functionals are related to the so-called ’excess’ functionals that arise from optimising the Dirichlet energy functional under certain boundary conditions (Bevan 14). The study of polyconvex functionals is of great interest in the area of elasticity (see Open Problems in Elasticity, Ball 02) and features many un- solved problems that seemingly do not yield to the standard methods in the field.
In particular, we will establish conditions for a certain parameterised family of polyconvex functionals to have a global minimum. The functionals we will be considering do not meet the usual conditions to be treated by the direct method in the calculus of variations and so we will make use of other techniques including numerical approximations, partial differential inclusions and Fourier- type analysis. We will develop several techniques for solving PDIs and evaluate the benefits of these methods in relation to analysing the functionals under consideration. Overall, we will obtain bounds on a parameter in both the cases of sufficient and necessary conditions for the existence of a global minimiser.
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Calculus in Infinite Dimensions
Taste of Research for Undergraduate Students
University of Surrey • February 2023
Have you ever wondered how to take a derivative in infinite dimensions? How about optimising a function with an uncountable number of variables? These questions and more can be addressed by the ’calculus of variations’, an area of analysis that establishes techniques to optimise functions by considering small changes of variables (called variations).
The talk shall begin with an introduction to the calculus of variations, summarising some of the key concepts and results, with particular focus on the contrast with ’standard’ calculus. We shall explore various optimisation tools, such as the Euler-Lagrange equation and the direct method, discussing the advantages and limitations of each. Here we will note the importance of the various definition of convexity, which will also go over.
We will finish with an overview of some results in current research. In particular, these results will relate to the minimisation of a parameterised family of polyconvex functionals that are related with some open problems in the area of elasticity. Here we will also introduce partial differential inclusions, a generalisation of partial differential inclusions, and the problem of potential wells.
Skills
Public Speaking
Gave presentations at conferences and seminars.
Education
Lead tutorials and labs for undergraduate mathematics courses.
Research
Experience conducting independent research in mathematical analysis.
Programming
Proficient in Python, MATLAB, and LaTeX for mathematical modeling and simulations.