Presentations
Minimisation of the Dirichlet Energy amongst a class of constrained maps
University of Surrey
Introduction to my Research • December 2024
Minimisation of a Constrained Energy Functional
University of Surrey
Highlights of 2024 • December 2024
Minimisation of the Dirichlet Energy amongst a class of constrained maps
University of Bath
Analysis Seminar • 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 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 ) 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 is varied. In particular, we will discuss methods for establishing sufficient 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 sufficient and necessary bounds on the parameter. Time allowing, we will then derive the maps , associated with these pressure functions , that minimise the Dirichlet energy subject to a pointwise Jacobian constraint.
Mean Hadamard Inequalities & Elasticity
Imperial College London
Junior Analysis Seminar • 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 here.
Optimisation of a 1D Family of Polyconvex Functionals
University of Bath
Singularities and Patterns in Evolution Equations • 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.
Optimisation of a 1D Family of Polyconvex Functionals
University of Oxford
SIAM Student Chapter Conference • 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.
Optimisation of a 1D Family of Polyconvex Functionals
University of Surrey
Internal Student Conference • 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.
Calculus in Infinite Dimensions
University of Surrey
Taste of Research for Undergraduate Students • 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.