*Published Paper*

**Inserted:** 30 may 2002

**Last Updated:** 23 jan 2003

**Pages:** 29-60

**Year:** 2002

**Notes:**

Nonlinear Problems in Mathematical Physics and related topics II, in honor of O.A. Ladyzhenskaya. M.S. Birman, S. Hildebrandt, V.A. Solonnikov and N.Uraltseva Eds., International Mathematical Series, Kluwer*Plenum.*

**Abstract:**

\documentclass{article}

\begin{document}

We estabilish a structure theorem for diverge-free vectorfields $u$ in $*R*^2$ arising in a micromagnetics model.
Precisely, we assume that $u$ is representable as $e^{i\,\phi}$ for a suitable bounded Borel
function $\phi$ and that the measure
$$
\mu_{\phi:=\int}_{{R}{\rm} div}e^{{i\,\min\{\phi,a\}}\,da
}
$$
is locally finite in $*R*^2$. We show that the $1$-dimensional part of $\mu_\phi$
(i.e. the set of points where the upper spherical $1$-dimensional density of $\mu_\phi$
is strictly positive) is countably rectifiable, and that out of this part $\phi$ has vanishing
mean oscillation. The proof is based on a delicate blow-up argument and on the classification
of all blow-ups.

\end{document}

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