Geometrically constrained walls |
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Authors: | Robert V Kohn Valeriy V Slastikov |
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Institution: | (1) Courant Institute, New York University, New York, NY 10012, USA;(2) Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK |
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Abstract: | We address the effect of extreme geometry on a non-convex variational problem. The analysis is motivated by recent investigations
of magnetic domain walls trapped by sharp thin necks. To capture the essential issues in the simplest possible setting, we
focus on a scalar variational problem with a symmetric double well potential, whose spatial domain is a dumbell with a sharp
neck. Our main results are (a) the existence of local minimizers representing geometrically constrained walls, and (b) an
asymptotic characterization of the wall profile. Our analysis uses methods similar to Γ-convergence; in particular, the wall
profile minimizes a certain “reduced problem” – the limit of the original problem, suitably rescaled near the neck. The structure
of the wall depends critically on the choice of scaling, specifically the ratio between length and width of the neck. |
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