Micromagnetism in the ultrathin limit

We derive some results concerning the static and dynamic micromagnetic behaviour of magnetic elements in the ultrathin limit. In this limit, a most remarkable, shape-independent, logarithmic correction of the magnetostatic energy appears, produced by the long-range character of the dipolar interaction. Apparently unrelated phenomena such as the pinning of the precessional motion at some of the boundaries of a thin magnetic element, the non-pinning at other boundaries and a multi-to-single-domain transition in ultrathin elements with perpendicular magnetization have a common origin: the logarithmic correction.