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Title:  Local minimizers in micromagnetics and related problems 
Authors:  Winter, M Ball, J Taheri, A 
Keywords:  Micromagnetics Implicit function theorem Calculus of variations EulerLagrange equation 
Publication Date:  2002 
Publisher:  Springer 
Citation:  Calc Var Partial Differential Equations 14: 127 
Abstract:  Let \Omega \subset {\bf R}^3 be a smooth bounded domain and consider
the energy functional
J_{\varepsilon} (m) := \int_{\Omega} \left( \frac{1}{2 \varepsilon}
Dm^2 + \psi(m) + \frac{1}{2} hm^2 \right) \, dx + \frac{1}{2}
\int_{{\bf R}^3} h_m^2 \, dx.
Here \varepsilon is a small non negative parameter and the space
of admissible
functions for m is the Sobolev space of vectorvalued functions
W^{1,2}(\Omega;{\bf R}^3) which satisfy the pointwise constraint
m(x)^21=0 for a.e. x \in \Omega.
The integrand \psi:{\bf S}^2
\to {\bf R} is assumed to be a sufficiently smooth non negative
density function with a multiwell structure. The function h_m
\in L^2 ({\bf R}^3; {\bf R}^3) is related to m via Maxwell´s
equations. Finally h \in {\bf R}^3 is a constant
vector. The energy functional J_{\varepsilon} arises from the study
of continuum models for ferromagnetic materials known as micromagnetics developed by W. Brown \cite{BR}.
In this paper we aim to construct local energy minimizers for this
functional. Our approach is based on studying the corresponding
EulerLagrange equation and proving a local existence result
for solutions close to a fixed constant solution. Our main device
for doing this is a suitable version of the implicit function theorem.
We then show that these solutions are local minimizers of
J_{\varepsilon} in appropriate topologies by using certain
sufficiency theorems for local minimizers.
Our analysis is applicable to a much broader class of functionals than
the ones introduced above and on the way of proving our main results
we reflect on some related problems. 
URI:  http://bura.brunel.ac.uk/handle/2438/520 
DOI:  http://dx.doi.org/10.1007/s005260100085 
Appears in Collections:  Mathematical Science Dept of Mathematics Research Papers

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