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Abstract

The topic of this thesis are magnetic domains in thin ferromagnetic films with strong perpendicular anisotropy. Our starting point is Micromagnetics, a continuum model based on the principle of minimal energy. At its core is the micromagnetic energy functional, whose local minimizer represent the stable magnetization configurations of the ferromagnetic body. Identifying a suitable thin film regime leads us to investigate a singular limit of the nonconvex and nonlocal micromagnetic energy functional. Our asymptotic analysis yields a scaling law for the typical domain size as a function of the film thickness and another material parameter. To prove an ansatz free lower bound of the energy, we extend an interpolation inequality first obtained in [26]. Moreover, we study a shape optimization problem that can be considered as a prototypical model for a single magnetic domain. We minimize the sum of the perimeter and the dipolar self-energy among subsets of R3 with prescribed volume. Upon proving that minimizers exist, we show that they are (L3-equivalent to) connected open sets with smooth boundary. We furthermore establish a scaling law for the minimal energy in terms of the prescribed volume which yields further information about the shape of minimizers.