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, KluwerPlenum.

Abstract:

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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.

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