Abstract
Recent advances in nanofabrication allow an unprecedented degree of control of ferromagnetic materials down to the atomic scale, resulting in novel nanostructures whose properties are often dominated by material interfaces. Mathematically, these systems give rise to challenging problems in the calculus of variations that feature non-convex, vectorial, topologically constrained, multi-scale variational problems. Yet despite the daunting complexity inherent in the problem arising from the 21st century technological applications, rigorous variational analysis can still elucidate energy-driven pattern formation in these systems. In this talk, I will discuss several examples of variational problems emerging from models of current ferromagnetic nanostructures under development. With the help of asymptotic techniques and explicit solutions, I will give three examples in which the energy minimizing configurations may be characterized in terms of optimal one-dimensional transition layer profiles separating magnetic domains with different magnetization orientation.