## Pattern Formation

### Domain Branching

The minimisers of variational problems involving the competition between short-range attractive and long-range repulsive interactions often exhibit quite intricate patterns. While proving the presence of particular patterns is often a very hard problem, a first step is to understand the scaling of the minimal energy in terms of the system parameters.

A prominent class of such problems arises as the sharp interface limit (in the sense of \(\Gamma\)-convergence) of diffuse interface variational problems, and is of the form $$ \text{minimise} \quad \mathcal{E}(u) = \mathcal{S}(u) + \mathcal{N}(u) $$ over an admissible class of functions \(u\) taking only the values \(\pm 1\), and \(\mathcal{S}\) is related to the interfacial energy between the two phases \(\{u=1\}\) and \(\{u=-1\}\), hence penalising small-scale phase variations.

I am working on problems relevant to micromagnetics , and on a closely related problem in the context of superconductivity . In both models, the nonlocal interaction \(\mathcal{N}\) favours small-scale phase oscillations to minimise the stray field energy outside of the sample (in micromagnetics language), and the minimiser is expected to exhibit domain branching towards the sample boundary, which should manifest itself through local energy estimates reflecting the scaling law of the minimiser and imply approximate (in a statistical sense) self-similarity. A related lower-dimensional model for twin-branching near an Austenite-Martensite interface in shape-memory alloys exhibiting this behaviour has been treated in the important works by Kohn-MÃ¼ller and Conti .