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On the rectifiability of defect measures arising in a micromagnetics model. (English) Zbl 1055.49008

Birman, Michael Sh. (ed.) et al., Nonlinear problems in mathematical physics and related topics II. In honour of Professor O. A. Ladyzhenskaya. New York, NY: Kluwer Academic Publishers (ISBN 0-306-47333-X/hbk; 0-306-47422-0). Int. Math. Ser., N.Y. 2, 29-60 (2002).
Summary: We establish a structure theorem for divergence-free vector fields \(u\) in \(\mathbb{R}^2\) arising in a micromagnetics model. Precisely, we assume that \(u\) is representable as \(e^{i\varphi}\) for a suitable bounded Borel function \(\varphi\) and that the measure \[ \mu_\varphi:=\int_R| \text{div}\, e^{i\min\{\varphi,a\}}|da \] is locally finite in \(\mathbb{R}^2\). We show that the one-dimensional part of \(\mu_\varphi\) (i.e., the set of points where the upper spherical one-dimensional density of \(u_\varphi\) is strictly positive) is countably rectifiable, and that out of this part \(\varphi\) has vanishing mean oscillation. The proof is based on a delicate blow-up argument and on the classification of all blow-ups.
For the entire collection see [Zbl 1005.00022].

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
78A99 General topics in optics and electromagnetic theory
35J20 Variational methods for second-order elliptic equations
35B25 Singular perturbations in context of PDEs
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