In this dissertation, I develop a phase-field approach to minimising a geometric energy functional in the class of connected structures confined to a small container. The functional under consideration is Willmore's energy, which depends on the mean curvature and area measure of a surface and thus allows for a formulation in terms of varifold geometry. In this setting, I prove existence of a minimiser and a very low level of regularity from simple energy bounds.
In the second part, I describe a phase-field approach to the minimisation problem and provide a sample implementation along with an algorithmic description to demonstrate that the technique can be applied in practice. The diffuse Willmore functional in this setting goes back to De Giorgi and the novel element of my approach is the design of a penalty term which can control a topological quantity of the varifold limit in terms of phase-field functions. Besides the design of this functional, I present new results on the convergence of phase-fields away from a lower-dimensional subset which are needed in the proof, but interesting in their own right for future applications. In particular, they give a quantitative justification for heuristically identifying the zero level set of a phase field with a sharp interface limit, along with a precise description of cases when this may be admissible only up to a small additional set.
The results are optimal in the sense that no further topological quantities can be controlled in this setting, as is also demonstrated. Besides independent geometric interest, the research is motivated by an application to certain biological membranes
