Variational models for pattern formation in biomembranes

Biological membranes are thin structures that are composed of various components. The different components often form microdomains that are arranged in complex
patterns. Several modeling approaches attribute many biological processes to these domains. To explain this pattern formation, variational models have been introduced that couple the local
composition of the membrane to its local curvature. Due to their nonlocal and nonconvex structure, the resulting variational problems are analytically and numerically challenging.

Objective: To explain the formation of different structures, we plan to determine the optimal scaling of the minimal energy in terms of the problem parameters. The more quantitative focus will
lie on the regime in which complex patterns are expected. We plan to combine analytical tools from the calculus of variations with numerical techniques from machine learning and reduced
basis methods.